The difference, Old and New? First of all the label New Testament, Should read Renewed Covenant, a simple word Reference check will show this.
Jesus laid down His Life to Pay the Death Penalty, for the breaking of the Blood Covenant established at Mt Sinai, He also paid the Price for all Sin, remember God wanted to Smoke Israel after the Golden Calf deal, Moses by virtue of expressing to God that the Nations would think that He was not able to save these People and that they were lead out to just be Killed, after which the Tent of Meeting was made with all the Sacrificial ordinances, with which none were able to remove Intentional Sin. also after the giving of the 10 Commandments, the People didn't want to hear the Voice of God any more, and in the Dialog between God and Moses, God says to Moses, IF ONLY THEY HAD THE HEART, to fear Me and keep My Commandments. the Difference is that By Faith we Believe, and the Holy Spirit within us, which is the Love of God, His Spirit, and the Law Written in our Hearts, that we willingly keep His Commandments, because we Love Him, knowing that He Loved us First. Many will say that we are being Legalistic, I call it being Obedient, but Keeping the rules and regulations and Commandments of Men is being Legalistic, Dogma. I do not understand why Paul is the definable Authority in the New Testament, What ever Happened to all the RED letter words that were supposed to Have been the Words of OUR Savior, and as you have said, Matthew 5 - 17 these were the First words of our Savior recorded through the Apostle- Matthew, Matthew is the Gospel about the King from Heaven, laying down the Rule for the Kingdom! Sorry I do get long winded!