How is the textual discrepancy between Exodus 20:8 and Romans 14:5 explained?

Jesus said that The Sabbath was made for man, not man for the Sabbath. This indicates that Jesus is in charge and that the law is a tool toward the ends of loving God and loving people.

There is nothing special about whatever day people take a Sabbath on. The Sabbath serves a purpose. That purpose is God's purpose. That purpose is honoring humans as sacred by allowing them to rest, and thus communicating to them that they are not a means to an end of accomplishing tasks.

The purpose is what is sacred. The day itself is not. There are other ways we honor people, and don't force them to engage in 7 day a week labor. Taking a Sabbath doesn't inherently mean you are honoring workers either. But the Sabbath is a hard boundary as far as executing the purpose of honoring workers as sacred, and not just robots that do things.

"Let every man be fully persuaded in his own mind." doesn't mean that everyone can decide what is right or wrong for themselves. It means don't force someone to accept a truth until that person is ready to be persuaded and willing to accept that truth.

Consider the context leading up to this verse:

Receive one who is weak in the faith, but not to disputes over doubtful things. For one believes he may eat all things, but he who is weak eats only vegetables. Let not him who eats despise him who does not eat, and let not him who does not eat judge him who eats; for God has received him. Who are you to judge another’s servant? To his own master he stands or falls. Indeed, he will be made to stand, for God is able to make him stand. One person esteems one day above another; another esteems every day alike. Let each be fully convinced in his own mind.
— Romans 14:1–5

Paul is talking about recent converts (or those interested in converting) from Paganism. Some are vegetarians, some regard every day as special, some believe physical pleasure is wrong, some … .

Paul's point is not about any of these details. But he is concerned that some of the Christians are more concerned about these details than about the central message of Christianity.

The new converts need to learn about Christianity first, and they will eventually be able to accept that eating meat is allowed, that physical pleasure can be freely enjoyed, that certain days are for certain purposes, etc.

But if they are forced to eat meat against their will before they have truly learned that it is acceptable to do so, they will feel uncomfortable, they will resist, and they will be distracted from learning the Gospel message. Forcing a vegetarian to eat meat, or ridiculing him for not doing so, is more likely to chase this person away than to teach him to be a Christian.

Paul is telling them to teach the essentials, and to allow time for the former Pagans to accept the things that they have spent their whole lives believing to be wrong. Insisting on immediate full acceptance of everything will only alienate them, and possibly lose them.

Teach the essentials of the Christian message, wait, and people will eventually be convinced by themselves to change their life-long views.

Paul does not deal with the question of Old vs. New covenants here. Rather, he is explaining that Christians should not judge each other over the issue of the Sabbath, among others. Paul's teaching here is not for or against Sabbath-keeping. Rather, he argues that Christians respect each others' view on the issue. The key sentences here are "Let every one be fully convinced in his own mind. He who observes the day, observes it in honor of the Lord."

Let's take a deeper look at the chapter:

As for the man who is weak in faith, welcome him, but not for disputes over opinions. 2 One believes he may eat anything, while the weak man eats only vegetables. 3 Let not him who eats despise him who abstains, and let not him who abstains pass judgment on him who eats; for God has welcomed him. 4 Who are you to pass judgment on the servant of another?... 5 One man esteems one day as better than another, while another man esteems all days alike. Let every one be fully convinced in his own mind. 6 He who observes the day, observes it in honor of the Lord. He also who eats, eats in honor of the Lord, since he gives thanks to God; while he who abstains, abstains in honor of the Lord and gives thanks to God.

Paul's attitude here is similar to what he expresses on the issue of of food sacrificed to idols In 1 Cor. 8:

Some, through being hitherto accustomed to idols, eat food as really offered to an idol; and their conscience, being weak, is defiled. 8 Food will not commend us to God. We are no worse off if we do not eat, and no better off if we do. 9 Only take care lest this liberty of yours somehow become a stumbling block to the weak. 10 For if any one sees you, a man of knowledge, at table in an idol’s temple, might he not be encouraged, if his conscience is weak, to eat food offered to idols?

For Paul, issues such as vegetarianism, Sabbath-keeping and eating food sacrificed to idols were not essential. Thus he says "Let every one be fully convinced in his own mind." The important thing is that people should follow their consciences, and they should avoid offending each other as much as possible.

Among the Pharisees [of whom Paul himself was one] there were often debates about minor points of Jewish law. Is it OK to eat without washing your hands first, or does this only refer to priests making offerings to God? Is it kosher to eat milk and meat together, or should we follow the literal letter of the law and simply refrain from cooking a kid in its mother's milk? Does keeping the Sabbath mean we must not even lift a hand to save another's life? Paul hated to see similar debates beginning to divide the church.

So how should the apparent discrepancy between the 10 Commandments and Paul's teaching to be explained? Not with reference to the Old vs the New Covenant. But rather as an expression of Paul's broad-minded interpretation of the Law, much as Reform Jews hold today. He does not speak against Sabbath-keeping. Indeed he supports it, saying that 'He who observes the day, observes it in honor of the Lord.'

The proper translation of Romans 14:5 is : One man esteemeth one day above another ; another esteemeth every day , meaning that some esteem every day as a perpetual state of rest in Christ, through faith.

ος μεν κρινει ημεραν παρ ημεραν ος δε κρινει πασαν ημεραν [Textus Receptus : Beza, Stephanus, Elzevir and Scrivener are identical : Undisputed]

one man esteems (a) day above (a) day ... another esteems every day.[Englishman's Greek New Testament, Interlinear, Literal Translation]

The words 'the same' or 'alike' (esteems 'every day the same'or 'every day alike') are not in the original, it is an interpolation and an interpretation.

Some highly regard one day above another ; some highly regard every day ; is the literal translation of the undisputed Greek text.

The sabbath was a figure. It was a foretaste, practically, of the rest that would be revealed once Christ came - the rest in faith that is in his work of redemption. The sabbath was but for a time, as a figure.

Now is come the true rest in Christ that is perpetual.

Once the true translation is realised and once the true experience of faith and a rest in faith is experienced, it is seen that there is no discrepancy in the texts mentioned.

One prefigures the other.

The second is the full realisation of the first.

This is reiterated by the writer to the Hebrews :

For we which have believed do enter into rest, as he said, As I have sworn in my wrath, if they shall enter into my rest: although the works were finished from the foundation of the world.

For he spake in a certain place of the seventh day on this wise, And God did rest the seventh day from all his works. 5And in this place again, If they shall enter into my rest. 6Seeing therefore it remaineth that some must enter therein, and they to whom it was first preached entered not in because of unbelief:

For he that is entered into his rest, he also hath ceased from his own works, as God did from his. 11Let us labour therefore to enter into that rest, lest any man fall after the same example of unbelief.

Hebrews 4: 4-6 and Hebrews 4: 10.

Without going into too great detail of a subject which is at the heart of the Gospel and, indeed, all of God's revelation to us, these two verses do not point up a discrepancy or contradiction but, instead, illuminate a difference between two covenants; one of law and the other of promise.

If there is considerable debate within Christendom concerning the role of the law (and there is) then the atheist is at a severe disadvantage in that they cannot ascribe promise keeping power to a deity they do not acknowledge.

The command (s) of the law, of which Sabbath keeping is one, given by God to Israel through Moses were not given to produce righteousness but, instead, were given to delineate and condemn unrighteousness:

Wherefore then serveth the law? It was added because of transgressions, till the seed should come to whom the promise was made; and it was ordained by angels in the hand of a mediator. Now a mediator is not a mediator of one, but God is one. Is the law then against the promises of God? God forbid: for if there had been a law given which could have given life, verily righteousness should have been by the law. - Galatians 3:19-21

Just as the law "Thou shalt not commit adultery..." was spoken to a humanity that is prone to it and, just as obeying this law outwardly does not remove the inner proclivity:

Ye have heard that it was said by them of old time, Thou shalt not commit adultery:  But I say unto you, That whosoever looketh on a woman to lust after her hath committed adultery with her already in his heart. - Matthew 5:27-28

so outward obedience to the Sabbath law does not remove the inner disposition to become unmindful of the Creator.

The Law, says the apostle Paul, was added because of transgressions [until the promised seed should come]. Paul (in 2 Corinthians 3) also refers to the law as a ministry of death and as a ministry of condemnation. He does so, not to point out a flaw in the law but to point out the flaw in man which the law was intended to restrain until the flaw in man is remediated. This is the Old Covenant. That schoolmaster or guardian kept us until the remediation occurred:

But the scripture hath concluded all under sin, that the promise by faith of Jesus Christ might be given to them that believe. But before faith came, we were kept under the law, shut up unto the faith which should afterwards be revealed. Wherefore the law was our schoolmaster to bring us unto Christ, that we might be justified by faith. But after that faith is come, we are no longer under a schoolmaster. For ye are all the children of God by faith in Christ Jesus. - Galatians 3:22-26

The New Covenant is that remediation. Jesus has represented humanity in the perfect keeping of the Old Covenant in that he positively never disobeyed either in word, deed, or intent (no sin was found in Him) and also in that he bore humanity's penalty for disobedience in Himself:

Who his own self bare our sins in his own body on the tree, that we, being dead to sins, should live unto righteousness: by whose stripes ye were healed. - 1 Peter 2:24

Once one comes to Christ there is no longer any need for the guardian/law and this is not because adultery is now permitted but because the Christian now has been given a new spirit ... the guardian is no longer external (law) but internal. It is actually the Spirit of the Living God, the Spirit of Christ, who lives within and teaches and enables the believer to keep the righteous requirement of the law:

There is therefore now no condemnation to them which are in Christ Jesus, who walk not after the flesh, but after the Spirit. For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death. For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh: That the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit. - Romans 8:1-4

This is why Paul is able to say "One man esteemeth one day above another: another esteemeth every day alike. Let every man be fully persuaded in his own mind." and that is clarified in the following verses:

He that regardeth the day, regardeth it unto the Lord; and he that regardeth not the day, to the Lord he doth not regard it. He that eateth, eateth to the Lord, for he giveth God thanks; and he that eateth not, to the Lord he eateth not, and giveth God thanks. For none of us liveth to himself, and no man dieth to himself. For whether we live, we live unto the Lord; and whether we die, we die unto the Lord: whether we live therefore, or die, we are the Lord's. For to this end Christ both died, and rose, and revived, that he might be Lord both of the dead and living. - Romans 14:6-9

As can be seen, under the new covenant the believer who keeps one particular Sabbath and the believer who considers every day as the Lord's have one thing in common ... they regard it unto the Lord. This is what the law is schoolmaster of and what Jesus has brought to pass:

And whatsoever ye do in word or deed, do all in the name of the Lord Jesus, giving thanks to God and the Father by him. - Colossians 3:17