How is Moses' and Jesus' contradictory teaching regarding oaths reconciled?

Jamieson, Fausset, Brown explain it this way:

But I say unto you, Swear not at all--That this was meant to condemn swearing of every kind and on every occasion--as the Society of Friends and some other ultra-moralists allege--is not for a moment to be thought. For even Jehovah is said once and again to have sworn by Himself; and our Lord certainly answered upon oath to a question put to Him by the high priest; and the apostle several times, and in the most solemn language, takes God to witness that he spoke and wrote the truth; and it is inconceivable that our Lord should here have quoted the precept about not forswearing ourselves, but performing to the Lord our oaths, only to give a precept of His own directly in the teeth of it. Evidently, it is swearing in common intercourse and on frivolous occasions that is here meant. Frivolous oaths were indeed severely condemned in the teaching of the times. But so narrow was the circle of them that a man might swear, says LIGHTFOOT, a hundred thousand times and yet not be guilty of vain swearing. Hardly anything was regarded as an oath if only the name of God were not in it; just as among ourselves, as TRENCH well remarks, a certain lingering reverence for the name of God leads to cutting off portions of His name, or uttering sounds nearly resembling it, or substituting the name of some heathen deity, in profane exclamations or asseverations. Against all this our Lord now speaks decisively; teaching His audience that every oath carries an appeal to God, whether named or not.

Matthew Henry says this.

II. It is here added, that the commandment does not only forbid false swearing, but all rash, unnecessary swearing: Swear not at all, v. 34; Compare Jam. 5:12. Not that all swearing is sinful; so far from that, if rightly done, it is a part of religious worship, and we in it give unto God the glory due to his name. See Deu. 6:13; 10:20; Isa. 45:23; Jer. 4:2. We find Paul confirming what he said by such solemnities (2 Co. 1:23), when there was a necessity for it. In swearing, we pawn the truth of something known, to confirm the truth of something doubtful or unknown; we appeal to a greater knowledge, to a higher court, and imprecate the vengeance of a righteous Judge, if we swear deceitfully. Now the mind of Christ in this matter is,

1. That we must not swear at all, but when we are duly called to it, and justice or charity to our brother, or respect to the commonwealth, make it necessary for the end of strife (Heb. 6:16), of which necessity the civil magistrate is ordinarily to be the judge. We may be sworn, but we must now swear; we may be adjured, and so obliged to it, but we must not thrust ourselves upon it for our own worldly advantage.
2. That we must not swear lightly and irreverently, in common discourse: it is a very great sin to make a ludicrous appeal to the glorious Majesty of heaven, which, being a sacred thing, ought always to be very serious: it is a gross profanation of God's holy name, and of one of the holy things which the children of Israel sanctify to the Lord: it is a sin that has no cloak, no excuse for it, and therefore a sign of a graceless heart, in which enmity to God reigns: Thine enemies take thy name in vain.
3. That we must in a special manner avoid promissory oaths, of which Christ more particularly speaks here, for they are oaths that are to be performed. The influence of an affirmative oath immediately ceases, when we have faithfully discovered the truth, and the whole truth; but a promissory oath binds so long, and may be so many ways broken, by the surprise as well as strength of a temptation, that it is not to be used but upon great necessity: the frequent requiring and using of oaths, is a reflection upon Christians, who should be of such acknowledged fidelity, as that their sober words should be as sacred as their solemn oaths.
4. That we must not swear by any other creature. It should seem there were some, who, in civility (as they thought) to the name of God, would not make use of that in swearing, but would swear by heaven or earth, etc. This Christ forbids here (v. 34) and shows that there is nothing we can swear by, but it is some way or other related to God, who is the Fountain of all beings, and therefore that it is as dangerous to swear by them, as it is to swear by God himself: it is the verity of the creature that is laid at stake; now that cannot be an instrument of testimony, but as it has regard to God, who is the summum verum-the chief Truth.

So, the idea of absolutely zero swearing (oath taking) is not intended, lest other contradictions more silly surface. What is meant is no frivolous swearing.

The key issues seem to be haste, pride, and folly. This is what Solomon says. He does not deny the propriety of making vows, just the wisdom of doing so. It is too easy to make a foolish vow that we in our pride think we can fulfill but often cannot.

Ecclesiastes 5

1 Guard your steps when you go to the house of God. Go near to listen rather than to offer the sacrifice of fools, who do not know that they do wrong.

2 Do not be quick with your mouth, do not be hasty in your heart to utter anything before God. God is in heaven and you are on earth, so let your words be few.

3 As a dream comes when there are many cares, so the speech of a fool when there are many words.

4 When you make a vow to God, do not delay in fulfilling it. He has no pleasure in fools; fulfill your vow.

5 It is better not to vow than to make a vow and not fulfill it.

6 Do not let your mouth lead you into sin. And do not protest to the [temple] messenger, "My vow was a mistake." Why should God be angry at what you say and destroy the work of your hands?

7 Much dreaming and many words are meaningless. Therefore stand in awe of God.

The whole chapter of Matthew 5 is full of hyperbolic language. Teachings like "whoever calls someone a fool, is liable to hell; lustfully looking is adultery; cutting off body organs is better than going to hell with them by committing sin; don't resist evil but turn the other cheek; love your enemies" etc. They aren't meant to be taken literally, he was teaching them the sense of the commandments. What is relatively better in relative terms. Just because there are provisions to take oath and diverse, it doesn't mean they should use them unnecessarily. Jesus was also showing he was greater than Moses.

The believers developed cunning laws of vows to overrule the commands of God. If a man gives the benefits for his parents as offering (to God) as a vow or oath, then he is freed or forbidden from all obligation towards them due to the vow. This is the actual meaning of taking the name of the Lord in vain. They had created ways to bypass and reject all commands through tradition or the twisting of the law. The similar cunning ways continued among sinners to bypass the law of God, even in the new covenant, so the sinner's nature has not changed.

[ESV Mark 7:6-13] ⁶And he said to them, “Well did Isaiah prophesy of you hypocrites, as it is written, “‘This people honors me with their lips, but their heart is far from me;⁷in vain do they worship me, teaching as doctrines the commandments of men.’⁸You leave the commandment of God and hold to the tradition of men.”⁹And he said to them, “You have a fine way of rejecting the commandment of God in order to establish your tradition!¹⁰For Moses said, ‘Honor your father and your mother’; and, ‘Whoever reviles father or mother must surely die.’¹¹But you say, ‘If a man tells his father or his mother, “Whatever you would have gained from me is Corban”’ (that is, given to God)—¹²then you no longer permit him to do anything for his father or mother,¹³thus making void the word of God by your tradition that you have handed down. And many such things you do.”

We read in the Nedarim many such provisions for vows that forbid others any benefit through the vows, and we can assume that various versions of such vows were allowed by the Pharisees.

Mishnah Nedarim commentary 8.7 [....] And, so too, in the case of one who says to another: Benefiting from me is konam for you, i.e., you are prohibited from deriving benefit from me, if you do not come and give my son one kor of wheat and two barrels of wine, Rabbi Meir says: It is prohibited for the other individual to benefit from the speaker until he gives the gifts to his son.[...]
Mishnah Nedarim 9.1 Rabbi Eliezer says: When halakhic authorities are approached with regard to the dissolution of a vow, they may broach dissolution with a person who took a vow by raising the issue of how taking the vow ultimately degraded the honor of his father and mother, asking him the following: Had you known that your parents would experience public shame due to your lax attitude toward your vow, would you still have taken the vow? But the Rabbis disagree with Rabbi Eliezer and prohibit broaching dissolution of a vow with this particular question. To support the opinion of the Rabbis, Rabbi Tzadok said: Instead of broaching dissolution with him by raising the issue of the honor of his father and mother, let them broach dissolution with him by raising the issue of the honor of the Omnipresent. They should point out that a vow taken in the name of God lessens the honor of God, so they could ask him: If you had known that your vow would diminish the honor of God, would you have taken your vow? And if so, if this is a valid method of broaching dissolution, there are no vows. Nevertheless, the Rabbis concede to Rabbi Eliezer with regard to a vow concerning a matter that is between him and his father and mother, that they may broach dissolution with him by raising the issue of the honor of his father and mother, as in this case the extenuation is connected to this particular vow.

Korban and Konam are words used in interchangeably as substitute words.

Ramban on Numbers 30:3:1 “[This means if] a person says: ‘May there be konam13The word konam is a substitute for korban , and is used for a vow of abstinence, meaning, “May this object be forbidden to me in the same way as it...disappears. for the Rabbis have said19Nedarim 2b. that in the case of vows one makes the [actual] object [referred to in the vow] forbidden to oneself [as when one says: “May this bread be konam13The word konam...is a substitute for korban , and is used for a vow of abstinence, meaning, “May this object be forbidden to me in the same way as it is forbidden to have any benefit from a holy offering.” upon me”], Commentary on the Torah by Ramban (Nachmanides). Translated and annotated by Charles B. Chavel. New York, Shilo Pub. House, 1971-1976

English Explanation of Mishnah Nedarim 1:2:2 One who says, “konam” “qonah” or “qonas”: these are the substitutes for korban. The normal substitute for “korban”, a sacrifice is “konam”. Our mishnah lists substitutes for “konam”. Mishnah Yomit by Dr. Joshua Kulp

It will be interesting to consider answering this question from other verses relating to Jesus teaching "differently" to what Moses taught.

He said to them, “Because your hearts were hard and stubborn Moses permitted you to divorce your wives; but from the beginning it has not been this way. Matthew 19:8 AMP

Here, we are able to derive or deduce that Moses' laws were not conclusive. They were simply out of the state of man at the time. and a careful study proves that, Jesus did not contradict the teaching. Moses said "if you are going to swear, at least do it in the name of God so you stay true to it" and Jesus says, "don't swear at all". He is talking about integrity here. He is saying "lets never get to the point we have to start swearing and doing things to prove our innocence. Just yes or no" In other words, the life of a Christian should be about being of high integrity.

No contradiction here.

Before Christ all sinned, and lived after the flesh, so God put laws in place to minimise suffering. He knew people were naturally sinful, and deceitful, so likewise to the Sabbath being a measure of control, that ‘holy day’ for the sinful, oaths were put into place being ‘holy words’ for the deceitful.

Now by Christ we can live after the Spirit not in the image of Adam, bearing only good fruits, knowing more intimately who God is & His will without the written law as an often unsuccessful middleman. In Christ there is no ‘more holy,’ ‘less sinful,’ there is only sainthood like our Heavenly Father. For this to occur we must have faith - a condition fulfillable by a child.

2 Corinthians 3:

‘Are we beginning to commend ourselves again? Or do we need, like some people, letters of recommendation to you or from you? You yourselves are our letter, written on our hearts, known and read by everyone. You show that you are a letter from Christ, the result of our ministry, written not with ink but with the Spirit of the living God, not on tablets of stone but on tablets of human hearts.

Such confidence we have through Christ before God, not that we are competent in ourselves to claim anything for ourselves, but our competence comes from God. He has made us competent as ministers of a new covenant — not of the letter but of the Spirit; for the letter kills, but the Spirit gives life.

Now if the ministry that brought death, which was engraved in letters on stone, came with glory, so that the Israelites could not look steadily at the face of Moses because of its glory, transitory though it was, will not the ministry of the Spirit be even more glorious? If the ministry that brought condemnation was glorious, how much more glorious is the ministry that brings righteousness! For what was glorious has no glory now in comparison with the surpassing glory. And if what was transitory came with glory, how much greater is the glory of that which lasts!

Therefore, since we have such a hope, we are very bold. We are not like Moses, who would put a veil over his face to prevent the Israelites from seeing the end of what was passing away. But their minds were made dull, for to this day the same veil remains when the old covenant is read. It has not been removed, because only in Christ is it taken away. Even to this day when Moses is read, a veil covers their hearts. But whenever anyone turns to the Lord, the veil is taken away. Now the Lord is the Spirit, and where the Spirit of the Lord is, there is freedom. And we all, who with unveiled faces contemplate the Lord’s glory, are being transformed into his image with ever-increasing glory, which comes from the Lord, who is the Spirit.’

Various creeds and confessions can be utilized in answering a question like this one. A general view has been that Jesus is denouncing or correcting actions that were abusing oaths and the inherent trust meant to be attached to them.

We should also take note of James's teaching in James 5:12 (ESV):

But above all, my brothers, do not swear, either by heaven or by earth or by any other oath, but let your “yes” be yes and your “no” be no, so that you may not fall under condemnation.

From the WCF 22 (with American Revision for 22.3):

1. A lawful oath is part of religious worship, (Deut. 10:20) wherein, upon just occasion, the person swearing solemnly calleth God to witness what he asserteth, or promiseth, and to judge him according to the truth or falsehood of what he sweareth. (Exod. 20:7, Lev. 19:12, 2 Cor. 1:23, 2 Chron. 6:22–23)
2. The name of God only is that by which men ought to swear, and therein it is to be used with all holy fear and reverence. (Deut. 6:13) Therefore, to swear vainly, or rashly, by that glorious and dreadful Name; or, to swear at all by any other thing, is sinful, and to be abhorred. (Exod. 20:7, Jer. 5:7, Matt. 5:34, 37, James 5:12) Yet, as in matters of weight and moment, an oath is warranted by the Word of God, under the new testament as well as under the old; (Heb. 6:16, 2 Cor. 1:23, Isa. 65:16) so a lawful oath, being imposed by lawful authority, in such matters, ought to be taken. (1 Kings 8:31, Neh. 13:25, Ezra 10:5)
3. Whosoever taketh an oath ought duly to consider the weightiness of so solemn an act, and therein to avouch nothing but what he is fully persuaded is the truth: (Exod. 20:7, Jer. 4:2) neither may any man bind himself by oath to any thing but what is good and just, and what he believeth so to be, and what he is able and resolved to perform. (Gen. 24:2–3, 5–6, 8–9)
4. An oath is to be taken in the plain and common sense of the words, without equivocation, or mental reservation. (Jer. 4:2, Ps. 24:4) It cannot oblige to sin; but in any thing not sinful, being taken, it binds to performance, although to a man’s own hurt. (1 Sam. 25:22, 32–34, Ps. 15:4) Nor is it to be violated, although made to heretics, or infidels. (Ezek. 17:16, 18–19, Josh. 9:18–19, 2 Sam. 21:1)
5. A vow is of the like nature with a promissory oath, and ought to be made with the like religious care, and to be performed with the like faithfulness. (Isa. 19:21, Eccl. 5:4–6, Ps. 61:8, Ps. 66:13–14)
6. It is not to be made to any creature, but to God alone: (Ps. 76:11, Jer. 44:25–26) and, that it may be accepted, it is to be made voluntarily, out of faith, and conscience of duty, in way of thankfulness for mercy received, or for the obtaining of what we want, whereby we more strictly bind ourselves to necessary duties: or, to other things, so far and so long as they may fitly conduce thereunto. (Deut. 23:21–23, Ps. 50:14, Gen. 28:20–22, 1 Sam. 1:11, Ps. 66:13–14, Ps. 132:2–5)
7. No man may vow to do any thing forbidden in the Word of God, or what would hinder any duty therein commanded, or which is not in his own power, and for the performance whereof he hath no promise of ability from God. (Acts 23:12, 14, Mark 6:26, Numb. 30:5, 8, 12–13) In which respects, popish monastical vows of perpetual single life, professed poverty, and regular obedience, are so far from being degrees of higher perfection, that they are superstitious and sinful snares, in which no Christian may entangle himself. (Matt. 19:11–12, 1 Cor. 7:2, 9, Eph. 4:28, 1 Pet. 4:2, 1 Cor. 7:23)

The Westminster Confession of Faith (Oak Harbor, WA: Logos Research Systems, Inc., 1996).

From the Heidelberg Catechism:

QUESTION 99. What is required in the third commandment?

ANSWER. That we must not by cursing, or by false swearing, nor yet by unnecessary oaths, profane or abuse the name of God; nor even by our silence and connivance be partakers of these horrible sins in others; and in sum, that we use the holy name of God no otherwise than with fear and reverence, so that he may be rightly confessed and worshiped by us, and be glorified in all our words and works.

Philip Schaff, The Creeds of Christendom, with a History and Critical Notes: The Evangelical Protestant Creeds, with Translations, vol. 3 (New York: Harper & Brothers, 1882), 343–344.

From the Thirty-Nine Articles of the Anglican Church

XXXIX Of a Christian Man’s Oath

As we confess that vain and rash Swearing is forbidden Christian men by our Lord Jesus Christ, and James his Apostle, so we judge, that Christian Religion doth not prohibit, but that a man may swear when the Magistrate requireth, in a cause of faith and charity, so it be done according to the Prophets’ teaching, in justice, judgment, and truth.

Philip Schaff, The Creeds of Christendom, with a History and Critical Notes: The Evangelical Protestant Creeds, with Translations, vol. 3 (New York: Harper & Brothers, 1882), 513–514.

Another helpful teaching can be found in J. I. Packer's Concise Theology:

(Note that he begins by citing Neh 5:12–13.)

Truth in relationships, especially between Christians, is divinely commanded (Eph. 4:25; Col. 3:9), and truth-telling is specified as integral to authentic godliness (Ps. 15:1–3)…

Oaths are solemn declarations that invoke God as a witness of one's statements and promises, inviting him to punish should one be lying. Scripture approves oath-taking as appropriate on solemn occasions (Gen. 24:1–9; Ezra 10:5; Neh. 5:12; cf. 2 Cor. 1:23; Heb. 6:13–17), though at the time of the Reformation the Anabaptists declined the practice as part of their rejection of involvement in the life of the secular world. They appeals to Jesus' condemnations of oaths devised and designed to deceive as if it were a rejection of oath-taking as such rather than a call for honest speech and a warning against the temptation to use words that give false impression, with manipulation and exploitation as one's real purpose (Matt. 5:33–37; cf. James 5:12).

Vows to God are the devotional equivalent of oaths and must be treated with equal seriousness (Deut. 21:23; Eccles. 5:4–6). What one swears or vows to do must at all costs be done (Ps. 15:4; cf. Josh. 9:15–18). God requires us to take seriously not only his words but our own as well. However, "no many may vow to do anything forbidden in the Word of God, or what would hinder any duty therein commanded" (Westminster Confession XXII.7).

J. I. Packer, Concise Theology: A Guide to Historic Christian Beliefs. (Carol Stream, IL: Tyndale House, 1993), 190–192.