I propose that a key difference is this: who is making the rules?
If we accept the OT concept of God as the Creator of the heavens and the earth--why can't He be legalistic? What basis would we have for challenging His authority? (didn't work out too well for those who tried, e.g. priests of Baal who opposed Elijah)
The OT speaks of a God who gives rules and commands and has every right to do so. One example (of many):
I am the LORD, I have no peer (Isaiah 45:5)
Jesus explicitly acknowledged He was not trying to destroy the laws given by God. From Matthew 5:
17 Think not that I am come to destroy the law, or the prophets: I am
not come to destroy, but to fulfil.
18 For verily I say unto you, Till heaven and earth pass, one jot or
one tittle shall in no wise pass from the law, till all be fulfilled.
19 Whosoever therefore shall break one of these least commandments,
and shall teach men so, he shall be called the least in the kingdom of
heaven: but whosoever shall do and teach them, the same shall be
called great in the kingdom of heaven.
Jesus (and Paul apparently) took great exception, on the other hand, to people who thought they could improve upon the law. From Mark 7:
5 Then the Pharisees and scribes asked him, Why walk not thy disciples
according to the tradition of the elders, but eat bread with
6 He answered and said unto them, Well hath Esaias prophesied of you
hypocrites, as it is written, This people honoureth me with their
lips, but their heart is far from me.
7 Howbeit in vain do they worship me, teaching for doctrines the
commandments of men.
8 For laying aside the commandment of God, ye hold the tradition
of men, as the washing of pots and cups: and many other such like
things ye do.
And in one of His most withering critiques of the scribes & Pharisees He pointed out their motivation:
But all their works they do for to be seen of men: they make broad
their phylacteries, and enlarge the borders of their garments,
Putting the pieces together
Paul's words have admittedly been stretched to justify many competing views, often in the absence of the context in which they were first stated. Gratefully, the Gospel authors provide a wealth of context for many of Jesus' teachings; upon reading His words I am left to conclude that all 3 of these statements are true:
God gives commandments and expects people to obey them (Matthew 7:21)
It is inappropriate for unauthorized people to try to expand upon God's rules and then judge others for not doing the same.
Obedience should not be about making oneself appear to be better than other people. And it looks like some of the legalistic-minded people criticized by Jesus and His apostles were doing just that.
The God of the Bible never abridges His own authority--He is the supreme law-giver. When His words are given through His appointed representatives they are authoritative and binding. When people without this authority try to make similar claims, they're speaking in the name of God without permission to do so. That is one of the most clear-cut ways to break the 3rd commandment:
Thou shalt not take the name of the Lord thy God in vain (Exodus 20:7)
Paul's concern is not to try to wriggle out of doing what God commands, but to tear down attempts to speak in the name of God and create rules & regulations never endorsed by God.