This short and succinct article below is extracted in full from : http://www.thenewcovenant.com/nc01.htm. There are also other enriching articles in this link that are worth pondering.
God said this New Covenant would be different, not like the Old Covenant (Jeremiah 31:31-32a). He also tells us that Jesus "is the mediator of a better covenant, which was enacted on better promises" (Hebrews 8:6). So the New Covenant is different from the Old Covenant, and better too. God tells us that "the Law came through Moses, but Grace and truth came through Jesus Christ" (John 1:17). Jesus didn't come to give us more law, He came to bring us His grace and truth.
Often, Christians are afraid of living a life which not based on law. They think that if a person does not base his life on keeping laws, his behavior will become "lawless." But if God controls your life, if His truth fills your thinking, you won't be lawless. This is the great difference between these two covenants, when it comes to how we live:
The Old Covenant relied on man's efforts to change his own behavior by obeying commandments, or law.
The New Covenant relies on Jesus living His own life in us, through His Spirit.
The New Covenant is not about our obedience to law. "For sin shall not rule over you, for you are not under Law, but under Grace " (Romans 6:14).
The New Covenant is about God's grace being multiplied in our lives. The New Covenant gives us the way to live, not by obeying law, but by trusting Christ to live His life through us. "If you walk in the Spirit, you will not fulfill the desires of the flesh." (Galatians 5:17)