The law is central to both.
The difference is that in the new, God said He will write it on in our hearts and minds.
Heb 8:10 For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people:
Knowing that the house of Israel is now anyone the believes in Jesus
Gal 3:28-29
28 There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Christ Jesus. 29 And if ye be Christ's, then are ye Abraham's seed, and heirs according to the promise.
Rom 2:26-29
26 Therefore if the uncircumcision keep the righteousness of the law, shall not his uncircumcision be counted for circumcision? 27 And shall not uncircumcision which is by nature, if it fulfil the law, judge thee, who by the letter and circumcision dost transgress the law? 28 For he is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh: 29 But he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter; whose praise is not of men, but of God.