The difference between the old and new covenant

Christianity sees itself as a continuation of what God was doing with the Jewish people. Christianity started due to Christ. Although it is seen this way, it did not start at the time of the incarnation or resurrection of Christ. We are the descendants of Abraham by faith. Christianity in a sense is at least as old as Abraham and therefore as old as Judaism. Further we see God's plan for a redeemer in Genesis 3 so the plan of Christianity dates from at least the Fall, and an argument can be made for the time of creation.

So how does the new covenant we have in Christ differ from the old covenant of Moses?

We live in the age of grace and not the age of the law, but this is not the difference. Paul states that the law was given to show the transgression. The Israelites were unable to keep it.

Now the law came in to increase the trespass (Romans 5:20)

Why then the law? It was added because of transgressions, until the offspring should come to whom the promise had been made (Galatians 3:19)

We live post resurrection but this is not the difference. We are saved thru the blood of Christ, but then so were the patriarchs. All men are saved thru Jesus, even those who lived before his death and resurrection. The blood of bulls and goats was unable to atone for sin (Hebrews 10:4). Jesus blood however can atone for any man:

For by a single offering he has perfected for all time those who are being sanctified. (Hebrews 10:14)

We live by faith but this is not the difference. Abraham was justified by faith and this before the Mosaic Law. Habakkuk tells us that the just shall live by faith (Habakkuk 2:4). Faith was the criteria for salvation even under the Mosaic Law. Of course since the time of Christ we are aware of who our salvation is thru.

The change is power. We have been given the Holy Spirit who gives us the ability to follow God. Both Jeremiah and Ezekiel spoke of this during the age of Law.

But this is the covenant that I will make with the house of Israel after those days, declares the LORD: I will put my law within them, and I will write it on their hearts. And I will be their God, and they shall be my people. And no longer shall each one teach his neighbor and each his brother, saying, 'Know the LORD,' for they shall all know me, from the least of them to the greatest, declares the LORD. For I will forgive their iniquity, and I will remember their sin no more." (Jeremiah 31:33)

And they shall be my people, and I will be their God. I will give them one heart and one way, that they may fear me forever, for their own good and the good of their children after them. I will make with them an everlasting covenant, that I will not turn away from doing good to them. And I will put the fear of me in their hearts, that they may not turn from me. (Jeremiah 32:38-40)

And I will give them one heart, and a new spirit I will put within them. I will remove the heart of stone from their flesh and give them a heart of flesh, that they may walk in my statutes and keep my rules and obey them. And they shall be my people, and I will be their God. (Ezekiel 11:19-20; see also 36:26-27).

Hebrews confirms this

This is the covenant that I will make with them
after those days, declares the Lord:
I will put my laws on their hearts,
and write them on their minds (Hebrews 10:16).

The new covenant is marked in time by the death and resurrection of Jesus. And this event is imperative to our salvation: both before and after the incarnation. This is the central event of the Christian faith. But the change in covenant is not the age of grace, nor the fact that salvation is thru Jesus, nor the necessity of faith: these were and continue to be fundamental to salvation. The new covenant is a new heart, an ability to know and live these things by the power of God's Spirit whom he gives without limit (John 3:34).