These two concepts are not at all the same in the way they work out in practice. I believe the new covenant is summed up in the words of Peter when he told us to take heed to the Scriptures until the day dawns and Christ, the Morning Star, arises in our heart.
This teaches us that if we are to bring forth Christ, we, in a sincere adamic personality, must read the Scriptures and pray for help in being obedient to the Word until Christ is formed in us. Such is the true approach to salvation, to the new creation.
Here are the traditional "four steps of salvation:
All have sinned and come short of the glory of God.
We cannot save ourselves.
Christ died to forgive our sins.
When we die we shall go to Heaven to live forever in a mansion.
There is no moral transformation, no new creation, in the traditional four steps of salvation. Therefore this is an unscriptural way of salvation.