Quoting the Catechism:

1750 The morality of human acts depends on:

• the object chosen;
• the end in view or the intention;
• the circumstances of the action.

The object, the intention, and the circumstances make up the "sources," or constitutive elements, of the morality of human acts.

That a person may choose between NFP and contraception to achieve exactly the same end is no objection to one path being morally licit and the other morally illicit, for exactly the same reason why my ability to choose between working extra hours and robbing a store to achieve exactly the same end, namely availing myself of some extra money, is no objection to one path being morally licit and the other morally illicit.

In both cases, the reason why one path is morally illicit is that the object of the act is intrinsically disordered. Quoting again the Catechism:

1756 It is therefore an error to judge the morality of human acts by considering only the intention that inspires them or the circumstances (environment, social pressure, duress or emergency, etc.) which supply their context. There are acts which, in and of themselves, independently of circumstances and intentions, are always gravely illicit by reason of their object; such as blasphemy and perjury, murder and adultery. One may not do evil so that good may result from it.

It is easy to see why NFP is not morally disordered according to its object: there is nothing wrong in the spouses abstaining from sex for some days by mutual agreement (1 Cor 7:5).

It may not be so easy to see why contraception is morally disordered according to its object, but if you are Catholic it is not actually necessary for you to see why it is so, as faith in the divine assistance to the Church's Magisterium is enough for you to know that it is so. Still, it is certainly good being able to understand why, and Pope Paul VI provides an explanation in HV 11-14, which I summarize below.

Just as, by design of God inscribed in human nature, marriage has two ends, namely "the procreation and education of offspring" and "the good of the spouses" [1], the marital act, by design of God inscribed in human nature, has two ends, namely the procreative and the unitive. "Hence to use this divine gift while depriving it, even if only partially, of its meaning and purpose, is equally repugnant to the nature of man and of woman, and is consequently in opposition to the plan of God and His holy will" (HV 13).

[1] Code of Canon Law, 1055.

Natural Family Planning (NFP) does not interfere with the marital act. Interfering with it is what caused God to punish Onan in Genesis 38:9-10.

NFP, also known as periodic abstinence, is supported by St. Paul when he writes (1 Corinthians 7:3-5):

3 Let the husband render the [marriage] debt to his wife, and the wife also in like manner to the husband.

4 The wife hath not power of her own body, but the husband. And in like manner the husband also hath not power of his own body, but the wife.

5 Defraud [deprive] not one another [of rendering the debt], except, perhaps, by consent, for a time, that you may give yourselves to prayer; and return together again, lest Satan tempt you for your incontinency.

I would like to add to the ltcomdata's excellent exposition of classic Catholic ethical theory with a couple of considerations.

TL;DR

The Church condemns contraception—that is, impeding the fecundity of an otherwise fertile sexual act—because it is harmful in various ways to the persons (particularly the married couples) who take part in it. In particular, it places a barrier to the total self-giving that spouses are called to, it dissociates the sexual act from its ultimate purpose (cooperating with God to bring new human beings into the world), and it tends to lead each spouse to view the other as an object of gratification. Moreover, it places a particularly harsh burden on women, who generally are the ones who must undergo treatment to prevent pregnancy.

Natural family planning, on the other hand, is a form of periodic abstinence, hence it is not a form of contraception at all. If done for the right reasons, it avoids all of the problems mentioned above and can even be beneficial for the unity of a marriage.

(Note that the Church does not take a firm position as to whether using contraceptives in an extramarital situation adds additional gravity to, say, sins of fornication or adultery.)

We will now look at each of these in more detail.

What does “contraception” mean?

A common misconception is that the Church condemns artificial contraception, but not “natural contraception” (which, as we will see, is a misnomer). In fact, the artificialness or naturalness of contraception has nothing to do with the Church’s condemnation of the practice.

Pope Paul VI in his landmark encyclical Humanae vitae (HV) defines contraception as

any action which either before, at the moment of, or after sexual intercourse, is specifically intended to prevent procreation—whether as an end or as a means (HV 14).

This echoes what Pope Pius XI wrote in his 1930 encyclical Casti connubii (CC):

any use whatsoever of matrimony exercised in such a way that the act is deliberately frustrated in its natural power to generate life is an offense against the law of God and of nature, and those who indulge in such are branded with the guilt of a grave sin (CC 56).

Interpreting some of the technical language, contraception, therefore, is defined as any act whose purpose is to render an otherwise fertile sexual act infertile. As can be seen, this can include both “artificial” methods, such as hormonal treatment or sterilization, as well as “natural” methods, such as interrupting the sexual act before it is completed.

Hence, there are three essential elements of an act of contraception:

• The couple must intend to engage in a sexual act.
• That act must be fertile (or at least judged to be fertile), if it weren’t for the act of contraception.
• Some action must be accomplished deliberately, so as to render it infertile.

To put it in simpler language, those who contracept illicitly seek the benefit of the sexual act, without, so to speak, being willing to pay the cost, which is the possibility of procreation.

Note that the evil of contraception does not stem from its end or purpose, which is the prevention or delay of conception. Rather, the evil stems from the means used to obtain that end; i.e., frustrating the fertility of an otherwise fecund sexual act.

How contraception differs from natural family planning

This definition helps us to see how contraception is very different from natural family planning. In contrast to contraception, in NFP,

• A decision is made to refrain from engaging in the marital act, at certain moments in the woman’s cycle.*
• The sexual acts that take place outside the time of ovulation are already infertile, and so they cannot be subject to contraception.
• No action is taken to render those acts infertile.

Hence, NFP fails all three conditions for contraception.

It should be noted that NFP brings with it at least two important advantages (in direct contrast to contraception): it fosters self-control, rather than indulgence; and it helps couples to communicate with each other regarding the marital act.

(I will mention in passing that natural family planning need not be a haphazard endeavor. Modern advances in gynecology make possible an accurate discovery of the woman’s fertile period, which can be used both to encourage and to prevent conception. See the Billings method and the Creighton model for more information.)

Why the Catholic Church condemns contraception

I have described what contraception is; it is now necessary to show why it is harmful.

Contraception impedes the complete self-giving of the spouses.

Married couples, as Genesis 2:24 suggests, are called to give themselves completely to one another, both spiritually and physically. The marital union involves the whole person of both spouses, but, as St. Paul points out, the physical union of the flesh is an important part of it:

For the wife does not have authority over her own body, but the husband does. Likewise the husband does not have authority over his own body, but the wife does (1 Cor. 7:4, ESV).

The fundamental problem with contraception is that it places a barrier to the complete self-giving that spouses are called to. As Pope John Paul II in his apostolic exhortation Familiaris consortio (FC) puts it,

[Couples who have recourse to contraception] act as “arbiters” the divine plan and they “manipulate” and degrade human sexuality—and with it themselves and their married partner—by altering its value of “total” self-giving. … This leads not only to a positive refusal to be open to life but also to a falsification of the inner truth of conjugal love, which is called upon to give itself in personal totality (FC 32, quoted in part in the Catechism of the Catholic Church, No. 2370).

(Anyone familiar with the writings of Karol Wojtyła—the future John Paul II—will recognize here the echoes of his “theology of the body.” Wojtyła likened sexuality to a kind of language that necessarily implies giving the entire self to the other without holding anything back. In effect, he argues, using contraception is, in a way, a type of lie: with words, the spouses profess their total and enduring love for one another, but by holding themselves back with contraception, their actions say something different. By using contraception, couples say with their actions, “I will continue using you for my benefit, but I am unwilling to accept the consequences of that use.” See the general audience that John Paul II gave on January 12, 1983, for more on this topic.)

One sign of the degradation in self-giving is that contraception can make it more difficult for spouses to communicate regarding the marital act (and hence more difficult to communicate in general), since one of the principal reasons for such communication—namely, the discussion regarding whether to have more children—is removed.

Note that, as I mentioned, natural family planning does not have this issue; rather, it tends to promote communication and cooperation, as the spouses must plan their marital acts together. There is no question here of abusing the sexual act for personal benefit, because abstaining from it requires self-control and self-denial for the sake of the other.

Contraception dissociates the sexual act from procreation

Closely related to the first problem with contraception is the fact that it disconnects sexuality from its raison d’être, which is procreation. In the language of Casti connubii,

Since, therefore, the conjugal act is destined primarily by nature for the begetting of children, those who in exercising it deliberately frustrate its natural power and purpose sin against nature and commit a deed which is shameful and intrinsically vicious (CC 54).

As can be seen, Pius XI subscribed to the theory by which procreation is the primary purpose of the sexual faculty; the Church does not insist on this theory, but she does hold the procreative end to be essential. In Humanae vitae’s language:

And if each of these essential qualities [of the marriage act], the unitive and the procreative, is preserved, the use of marriage [i.e., the marital act] fully retains its sense of true mutual love and its ordination to the supreme responsibility of parenthood to which man is called (no. 12).

Since contraception effectively suppresses the procreative aspect of the sexual act, a number of important consequences follow:

• The marital act, because it is now accessible essentially without sacrifice, also becomes less attractive and less pleasurable for the couple. (For a concise philosophical treatment on the tendency of sensual pleasure to become “saturated,” see R. Lucas Lucas, Man Incarnate Spirit: a Philosophy of Man Compendium, Circle Press 2005, pp. 218-219.)
• Breaking the link to procreation removes one of the fundamental restraints and opportunities for self-control in the the use of the marital act. That link encourages couples the think about and discuss whether they wish to accept the possibility of having another child. If they opt to delay or stop pregnancies altogether, they must make the sacrifice of abstaining for a few days a month. If, on the other hand, they decide to proceed to a pregnancy, they accept all of the joys and challenges of childbearing. Breaking the link to procreation removes these positive opportunities for selfless love.
• As a consequence, contraception tends to make couples view each other increasingly as objects or as means to obtain the benefit of the marital act—which is never a positive development.

Contraception has a number of bad consequences in society

Because using contraception tends to lead to the objectification of sexuality, it has a number of consequences in society. Paul VI in Humanae vitae 17 points out three of them:

• It can lead to marital infidelity and in general to sexual immorality, because it removes a natural restraint on sexual indulgence, placed there by its Creator:

  Let them first consider how easily this course of action [i.e., using contraception] could open wide the way for marital infidelity and a general lowering of moral standards. Not much experience is needed to be fully aware of human weakness and to understand that human beings—and especially the young, who are so exposed to temptation—need incentives to keep the moral law, and it is an evil thing to make it easy for them to break that law.

• It can lead to discrimination against and mistreatment of women:

  Another effect that gives cause for alarm is that a man who grows accustomed to the use of contraceptive methods may forget the reverence due to a woman, and, disregarding her physical and emotional equilibrium, reduce her to being a mere instrument for the satisfaction of his own desires, no longer considering her as his partner whom he should surround with care and affection.

• It can lead to unjust and burdensome intrusions by the state into the family:

  Finally, careful consideration should be given to the danger of this power passing into the hands of those public authorities who care little for the precepts of the moral law. … It could well happen, therefore, that when people, either individually or in family or social life, experience the inherent difficulties of the divine law and are determined to avoid them, they may give into the hands of public authorities the power to intervene in the most personal and intimate responsibility of husband and wife.

(It is a confirmation of Pope Paul VI’s teaching that every one of these evils has come to pass, to varying degrees, in the decades since Humanae vitae was published.)

We could also add that the widespread use of contraception, according to Pope John Paul II in his encyclical Evangelium vitae (EV), leads to a cultural attitude—a “contraceptive mentality”—that sees children as a “burden” rather than as a gift. Such an attitude greatly facilitates the acceptance of the far greater evil of abortion:

It may be that many people use contraception with a view to excluding the subsequent temptation of abortion. But the negative values inherent in the “contraceptive mentality”—which is very different from responsible parenthood, lived in respect for the full truth of the conjugal act—are such that they in fact strengthen this temptation when an unwanted life is conceived. Indeed, the pro-abortion culture is especially strong precisely where the Church's teaching on contraception is rejected (EV 13).

Conclusion

Hence, to summarize the answer to the O.P.'s question:

• Contraception is defined as an action that impedes the fecundity of an otherwise fertile sexual act.
• Although the fertility of sexual acts may not be frustrated, married couples may make use of the marital act even when those acts are infertile (through no fault of their own, evidently)—indeed such marital acts are to be encouraged. (See. e.g., 1 Cor. 7:5.)
• Natural family planning consists fundamentally in abstaining from marital relations. It does nothing to impede the fertility of sexual acts.
• The problem with contraception has nothing to do with its end (the delay or prevention of conception), but with the fact that it tends to make one’s husband or wife a means to obtain gratification, rather than a person to be loved for his own sake.
• Contraception causes a lot of harm: to the the individuals who practice it, to married couples, and to society.
• Natural family planning, when used for the right motivations, has none of the moral problems associated with contraception, and can even be helpful for strengthening a marriage. It encourages self-control and selfless love, whereas contraception encourages indulgence and self-love.

* It should be noted that the evil of contraception depends entirely on deliberate choice of the couple to suppress the fertility of their sexual acts. In more technical terms, the suppression of fertility must be voluntary. Hence, for instance, a woman who is undergoing hormone therapy for medical reasons—for example, to treat Polycystic Ovary Syndrome (PCOS)—will also be infertile as long as she is receiving the treatment. Since the motivation for taking the hormones is medical, not contraceptive, the suppression of fertility in this case is involuntary. It is, therefore, morally licit for a woman in this situation to engage in the sexual act with her husband.

I will try to answer the question entirely from a Catholic perspective, as that seems to be what the question demands.

The classic determination of a moral act has three parts (CCC 1749-1761):

1. Is the act intrinsically immoral --- which is to say, wrong in every circumstance --- by its very nature?
2. Is the intention good, or bad?
3. Would the circumstances surrounding the moral act make it have an evil result?

A moral act (which is to say, an act in which a human has the freedom to act one way or another) must pass those 3 tests above to make it a moral action.

Now, let's compare non-abortifacient artificial contraception and Natural Family Planning in regards to the tests above. I add the proviso that the artificial contraception be non-abortifacient because otherwise we are talking about abortion (even if early abortion), and such act is fairly equated to murder of the innocent (CCC 2270-2275): an intrinsic evil of its own. It goes without saying that we are assuming any sexual intercourse takes place within marriage in both cases, or else we are talking about fornication, which immediately fails the third requirement (CCC 2353).

In what follows, see (CCC 2366-2370)

Non-abortifacient artificial contraception fails the first test. By its very nature artificial contraception (even when it doesn't kill the unborn) is a contradictory act. On the one hand, the sexual act is supposed to be (among other things) a total self-giving of spouses to each other, and on the other hand, the contraceptive symbolism (and intended effect) is to withhold from each other an integral part of themselves: their fecundity --- whether abundant or slim. In a sense it is very much like deception. One would have strong objections to a man pretending to be someone else in order to marry a woman (and vice-verse) --- it would go against the very nature of marriage that requires honesty and trust.

On the other hand, Natural Family Planning (refraining from having sexual intimacy at times) does not fail this test. There is no moral requirement that a couple have sex every waking moment of their lives. And in fact, there are circumstances when they must not have sexual intimacy (for instance, after surgery in their nether regions). And engaging in Natural Family Planning does not engage in the double-speak that contraceptive sex does: it is the difference between lying and remaining silent (See esp. CCC 2370). While we are called to always tell the truth, we are not always called to speak.

Of course, we are sometimes called to speak, and tell the truth. Analogously, God calls us to "be fruitful and multiply". Here is where we talk about intentions (the second requirement), which gets at the heart of the question. After all, aren't both methods (non-abortifacient artificial contraception and Natural Family Planning) intending to avoid children? And isn't that against God's will for humans stamped in their reproductive nature --- whichever method you use?

To be sure, there is much truth in these questions. If there was not the positive call from God to reproduce, one might be able to respond that both methods have the same ends, but one uses intrinsically flawed means of accomplishment, while the other uses entirely natural means. That difference is not trivial, of course, and would suffice as an answer if there was not the positive command from God to "be fruitful and multiply". However, we are called to reproduce. For now, let me say this: if someone's use of Natural Family Planning has as its goal the avoidance of children for the sake of not having any children, then that person is clearly doing violence to one of the purposes of matrimony (which is to obey the command to "multiply"), and as such has failed the second requirement of a moral act by having the wrong intention. Which is to say, Natural Family Planning is not "intrinsically moral" or "intrinsically good" (always right).

But how can it be right at all? Isn't the command to be "fruitful and multiply" definitive and absolute? Shouldn't the mandate be to find the best time for procreation every month and make sure to have sex during those times (whether or not on other times as well)?

Well, but we have been given other mandates from God on the subject of marriage. One of them is, for example, to care well for the children we do have. In other words, the command of God to "be fruitful and multiply" would not be fulfilled very well at all if we left our children to fend for themselves once we gave them birth. Part of the command is that we must do our best to make sure our children have a good chance of surviving in this world. And that may mean that we must have less children than we are physically able to have in order to be able to feed and care for the ones we do have.

Among other things, we are also called to respect our spouses bodies, for instance, and we must not engage in sexual intimacy when our wife is likely to be fecund if she is physically unable to handle the pregnancy.

In other words, we must be responsible with the gift of fecundity we are given (CCC 2367-2368), and take into account all other commandments from God. But we must have the right intentions. An avoidance of children must be motivated not by the wish not to have to sacrifice --- we are all called to sacrifice, just as Jesus sacrificed Himself --- our affluent lives, or by the wish to make all procreation impossible, or by selfish reasons, but must instead be "in conformity with the generosity appropriate to responsible parenthood" (CCC 2370). Our parenthood must be both responsible and generous, and never selfish. To take one instance, one must not forgo having a child because one really wants a boat.

Okay, but isn't it possible to have this lofty (and very worthy) aims in mind when one uses non-abortifacient artificial contraception to avoid children? Yes! In regards to intentions, the intention to avoid children is the same whether one uses non-abortifacient artificial contraception or Natural Family Planing, and in both cases the intentions can be worthy and moral, or immoral. However, assuming morally acceptable intentions, the question of means now shows up again. Whereas it is quite natural to use Natural Family Planning to space one's children, non-abortifacient artificial contraception is intrinsically evil because, by its very nature, it lies in a very fundamental way which is entirely contradictory to the essence of matrimony.

Finally, circumstances. Some actions which may be otherwise moral, may end up becoming immoral depending on the circumstances. A classic example of this might be the following: while it is morally good to attend Church on Sundays, it might not be moral to do so while in a foreign country that actively persecutes Christians if by doing so the attention of the authorities would be called to an underground group of Christians in hiding. In other words, consequences of an action are evaluated here. I have already included some language of circumstances above in my discussion of intentions. As far as Natural Family Planning goes, its morality might change or might be different if talking about a well-to-do family compared to a poor family, or if talking about a young and healthy couple compared to an elderly or not so healthy couple, or if talking about a large family compared to a small family. The devil here is in the details (and the moralist). In any case, the circumstances do not ever change the nature of an intrinsically evil act to a morally acceptable one (CCC 1754). They might lessen the moral culpability of an evil act, but the action remains as evil as ever. They might also increase the moral culpability of an evil act (lying about one's finances is less problematic when doing it to your friend than to the court, for instance). Similarly, circumstances can also increase or diminish the moral worthiness of a moral act (volunteering for the army would be more courageous in times of war than in times of peace, for instance). In any case, there are no circumstances in which non-abortifacient artificial contraception becomes a good act (that is the meaning of intrinsic evil, after all).

Well, that was too long. I wonder how many will read it all the way through?