Old Covenant
(Law of Moses)
“For the law was given by Moses,” John 1:17

New Covenant
(Grace)
“[but] grace and truth came by Jesus Christ.” John 1:17b

You can sum up the requirements of the Old Covenant with this thought.  
“Do this and thou shalt live.  Don’t do this and thou shalt die.”
(Ex. 19:5,  Ex. 31:12,  Deut. 30:16-20  Rom. 10:5)

You can sum up the requirements of the New Covenant with this thought.  
“Believe this and you shall live.  Don’t believe this and you shall die.”  
(Rom. 10:6-11,  John 5:24, 8:24)

The Law of Moses was introduced under the Old Covenant.
(Ex. 19:1-20:26,  Heb. 7:11,  John 1:17)

Grace and Mercy was introduced under the New Covenant.  (John 1:16-17, Heb. 8:12, 10:16,17)

The Old Covenant demanded perfect obedience to God’s holy law.  A transgression on just one point brought the penalty of the law which was death.  (Rom. 6:23, Gal. 3:10-12, James 2:10)

The New Covenant demanded faith in Christ.  By placing their faith in Christ Jesus a person would become righteous and receive life.  (Acts 13:37-39, Rom. 3:21-22,  John 1:12, 3:16, Eph. 2:7-9)

The Old Covenant was one of condemnation.  There was no mercy or grace under the law of the Old Covenant. The Old Covenant was external written on stone.  It brought forth death and condemnation. (2 Cor. 3:6,7,9,  Gal. 3:10,11)

The New Covenant was one of grace and mercy.  It was internal written on the heart.  It brought forth life and peace with God.  (John 1:16, Eph. 2:7-9,  2Cor. 3:3,6, Heb. 8:10, 10:16)

The Old Covenant had a fault in that it could not produce life or bring one near to God.  (Heb. 8:7,8, Heb. 7:18,19, 10:1-4, Rom. 3:20, Gal. 3:21-25) 

The New Covenant had the power to give life.  We draw near to God through the New Covenant.   (John 1:12,  Heb. 7:19,  9:9-15)

The Old Covenant was abolished, annulled and made obsolete by the New Covenant.  It was temporal and made of no effect by the New Covenant.  (Rom. 7:1-6, Eph. 2:13-16, Heb. 7:18,19  8:13, 10:9)

The New Covenant is everlasting.  It is based on the promises of God.(Heb. 13:20, Is. 55:3, 61:8)

The Old Covenant was given to the house of Israel only.  It never was given to the Gentile people.  It became void with the bringing in of the New Covenant. (Ex. 19:1-6,  Heb. 7:18, 8:13, 10:9)

The New Covenant was a better Covenant with better promises.  It was given to the Jew as well as the Gentile. (Heb. 7:22, 8:6, Eph. 2:11-22, Acts 28:28)

No flesh was ever saved or justified by the law.
Deut. 27:26,  Acts 13:37-39, Rom. 3:20, Gal. 3:10-12, James 2:10

Old Covenants saints were redeemed as a result of the New Covenant.  (Heb. 9:15, 11:7-40)