What then? shall we sin, because we are not under the law, but under grace? God forbid.
Know ye not, that to whom ye yield yourselves servants to obey, his servants ye are to whom ye obey; whether of sin unto death, or of obedience unto righteousness?
But God be thanked, that ye were the servants of sin, but ye have obeyed from the heart that form of doctrine which was delivered you.
Romans 6:15-17
You'll notice that Paul is talking here about obedience and how the obedience of those that were under the law has gained them life (obedience unto righteousness). If he is saying "do this and live" or "do this and you will be righteous" then there is no difference between his message and the message of Sinai. Yet this same Paul said this :
Tell me, ye that desire to be under the law, do ye not hear the law?
For it is written, that Abraham had two sons, the one by a bondmaid, the other by a freewoman.
But he who was of the bondwoman was born after the flesh; but he of the freewoman was by promise.
Which things are an allegory: for these are the two covenants; the one from the mount Sinai, which gendereth to bondage, which is Agar.
For this Agar is mount Sinai in Arabia, and answereth to Jerusalem which now is, and is in bondage with her children.
But Jerusalem which is above is free, which is the mother of us all.
For it is written, Rejoice, thou barren that bearest not; break forth and cry, thou that travailest not: for the desolate hath many more children than she which hath an husband.
Now we, brethren, as Isaac was, are the children of promise.
But as then he that was born after the flesh persecuted him that was born after the Spirit, even so it is now.
Nevertheless what saith the scripture? Cast out the bondwoman and her son: for the son of the bondwoman shall not be heir with the son of the freewoman.
So then, brethren, we are not children of the bondwoman, but of the free.
Galatians 4:21-31
Sinai and the law is here in view - and the covenant that went with it. Paul makes it clear that that covenant is not consonant with the new covenant in Christ's blood. So when he says this :
This I say then, Walk in the Spirit, and ye shall not fulfil the lust of the flesh.
For the flesh lusteth against the Spirit, and the Spirit against the flesh: and these are contrary the one to the other: so that ye cannot do the things that ye would.
Galatians 5:16-17
...he is using the same terminology he just finished using at the end of Galatians 4 above. The one born of the flesh and the one born of the Spirit are two distinct people and that is where the difference lies in terms of belonging to God. The one tries to please God by his own obedience but the other walks in that which God has prepared for him (Ephesians 2:10). Otherwise, why not just say this (instead of what he did write in Galatians 5:16) :
Obey what you are told and you shall not fulfill the lust of the flesh