The primary distinctions between the Old Covenant and the New Covenant of Jesus are the imputing of God’s righteousness to our account, the death of our “old man” and the indwelling of the Holy Spirit.
The Old Covenant actually was a covenant of Grace and faith.  God was gracious to His people just as much as He was in the New Covenant.  They could only enter into heaven by faith in God and the forgiveness of sins.  Romans 4 talks about this.  Abraham, before the establishment of the law, received imputed righteousness because of his faith.  David was forgiven of his sins and received imputed righteousness after the law, but it was by faith in God.  Israel was given a sacrificial system to point them to the coming Lamb of God.  Romans says that they put faith in God that their God ordained sacrifices would cover their sins.  God, in His forbearance (His waiting), counted the blood of their sacrifices until the Lamb of God could be sacrificed.  At the moment of Christs death, all their sacrifices were cashed in so to speak.  Their sacrifices were prophetic pictures of their faith in the Lamb to come and when He died on the cross He took all their prophetic acts and counted them as faith in His work on the cross.  They were only saved by faith in God that His system of sacrifices could wash away their sins.  The Old Covenant was never preached a salvation by works.  It is the same in the New Covenant except the Lamb of God has already come and we only need to receive His work to receive His righteousness.  We no longer have to wait for the final sacrifice in our day because Jesus has already finished His work.  Imputed righteousness means God has removed sin from our account and instead He has put in the righteousness of God in our account.  In a legal way, He looks at us as if we had the very innocence of Jesus Himself.

In the New Covenant our “old man”, or our corrupt nature is killed.  This refers to our dead spirit that was corrupt, dead to God, filled with darkness and void of the life, wisdom or destiny of God.  Our spirit is reborn, regenerated or recreated into a new man.  This new man (our spirit) is dead to sin and alive to God.  The new man consists of our resurrected spirit, the indwelling Spirit and the renewed mind (renewing is an ongoing process). Sin, dark thoughts, dark emotions and condemnation before God no longer have dominion over us.  We are no longer absolute slaves to them but can overcome them.  Our spirits are radically different now.  We can now live by our spirit like Adam and Eve lived by their spirit.  Before, we could only live by the senses and desires of our bodies and our souls.  Now we can live by the senses and desires of our spirit.  Living by our spirit is a growing process.

The last distinction is the indwelling of the Holy Spirit in our spirit (new man).  Jeremiah 31 and Ezekiel 36 describe the New Covenant that was to come by Jesus.  God spoke through them and said He would give His people are new heart, a new spirit, He would put His Spirit in them to cause them to walk in His ways and He would write His word upon their hearts and minds.  This is a huge distinction of the New Covenant and it is referenced all over the New Testament.  The Holy Spirit now lives in the spirit of a believer and that believer has the uncreated power of God in them to resist sin, be transformed supernaturally into the image (character) of Jesus, power to minister and walk in God’s wisdom and in the end to be raised by the dead by the Spirit.  Romans 8 and 2 Corinthians 3 are chapters focused on the New Ministry of the Spirit.  He will write the word of God on our emotions and our thoughts and radically change us on the inside.  2 Corinthians 3 says the Spirit will transform us from glory to glory as we receive the revelation of God from Him.  He will transform our inner man into the image of Jesus day by day, step by step and from glory to glory as we set our minds on Him.

The “letter of the law” alone will not give us power to obey God.  We must have the letter and the Spirit because the Spirit imparts life to us to obey God.  The laws of God are good they just don’t impart power to us to obey them, the indwelling Spirit does.  The letter of the law alone condemned because it gave no inner power to obey the law, it brought forth guilt of sin and legal condemnation before God.  But now we have been justified by Jesus’ sacrifice, received a new spirit and the Holy Spirit.  The Spirit imparts life to obey the law of God.  Believers are no longer condemned (guilty in God’s court) because they have been justified (legal innocence) by Jesus’ work on the cross.  We should still feel guilt when we are in disagreement with God’s words and should repent when we see it but now we can cultivate more and more godly desires and have power by the Spirit to walk out His law (be transformed into His loving image) and love instead.  Romans 13 and 1 Corinthians 13 describe the practical outworkings of love, which are the 10 commandments and the sermon on the mount.  Loving God and loving our neighbor mean walking out God’s commandments in the spirit of love.  We can now do that at the heart level because of the Spirit within us.

We access this new life by studying the benefits of our new man and the indwelling spirit, which renews our minds, then we literally set our mind and imagination on the indwelling Spirit (Rom 8) and we talk to Him.  Thank Him for all the benefits you read about and ask Him to release His power in you.