I took this from a web site. It is my understanding. This explains better than I can.
http://www.seekgod.ca/hr/hrfaqs1.htm
In the second chapter of Corinthians we see where Paul lists three differences between these two covenants.
2Corinthians 3:3
Forasmuch as ye are manifestly declared to be the epistle of Christ ministered by us, written not with ink, but with the Spirit of the living God; not in tables of stone, but in fleshy tables of the heart.
...6 Who also hath made us able ministers of the new testament; not of the letter, but of the spirit: for the letter killeth, but the spirit giveth life.
7 But if the ministration of death, written
and engraven in stones, was glorious, so that the children of Israel could not stedfastly behold the face of Moses for the glory of his countenance; which
glory was to be done away:
8 How shall not the ministration of the spirit be rather glorious?
9 For if the ministration of condemnation
be glory, much more doth the ministration of righteousness exceed in glory.
10 For even that which was made glorious had no glory in this respect, by reason of the glory that excelleth.
11 For if that which is done away
was glorious, much more that which remaineth
is glorious.
What Paul was saying was that the Old Covenant which was engraved in stones was glorious but that the ministry of the Spirit was more glorious, because the glory of the ministry of the law given Moses was passing away .The glory of the New overshadows the glory of the old. Thus we see that the first was a temporary one whose glory passed away, yet the New being eternal, remains.
The Old was a ministry of condemnation (v9) and death (v7), the New is the ministry of life(v6) and righteousness(v9).The law is holy but due to man´s inability to adhere to it, it became a ministry of death, showing us that man can never be righteous by his own efforts. As Isaiah the prophet said, all our righteousness is as filthy rags. Hence the need for God to provide the only way to Him, based on the righteousness of Christ.
It has been suggested that some who propagate the faulty scholarship regarding renew do so because if it is called a renewed covenant, they can then justify telling their adherents that they have to obey the Torah of Moses given at Sinai. Those who don't agree are considered non-believers and lost.
Some posit that God would never break His promise, so the Mosaic law or old covenant is still in effect. It is true that God cannot break His Word, which is what sealed the covenant with Abraham, Moses and for all other covenants. But people do break their word and promises, and the Scriptures do show that Israel was not faithful in keeping the Mosaic covenant. God repeatedly prefaced His instructions with "if" you do this, or "if" you don't.
God showed
He could change the covenant because of their disobedience--and He broke it so that a better way would be known.