When we say that the law of Christ is the New Covenant era version of the law of God we are saying that we are not under the Old Covenant era version of the law of God, the Mosaic Law. These two eras of law are also two different eras in terms of emphasis. In the Old Covenant era Israel was the temporary, unbelieving, picture of the people of God (Hebrews 8:7-8). When God gave them his law on Mount Sinai he gave it Israel so that their sin would increase (Romans 5:20, Romans 7:5). It was through their personal sin that Israel might see that they were in a hopeless position and that any true salvation must be all of God and by grace alone. In saying this it should be stated that God’s plan for Israel was that they would not believe, except for a remnant (Isaiah 6:9-13). In the Old Covenant the emphasis for all to see was law. The Mosaic law is given to us in rather neat lists (ex. Leviticus 19). This is due to the fact that Israel under the Old Covenant was unbelieving and the emphasis of law was the appropriate emphasis for a works covenant and an unbelieving people.
But when we come to the law of Christ in the New Covenant era we do not find the priority of law in the teaching passages. Why is this so? I would say that this is so because in the New Covenant era the issue of primary importance is not law but the new heart. The law of Christ has relevance in that it is in the keeping of the law that we show that we have a new heart (1 John 5:3). But it is the new heart that provides the motivation to keep the law of Christ (Romans 8:5-14, Romans 7:6). It is also the new heart which is the visible evidence that Jesus died for our sins and that we are the people of God (1 John 3:9-11. This is why the teaching passages of the New Covenant era have a different flavor than that of the Old Covenant era.
In the Old Covenant era the motivation was external, the Mosaic law, whereas in the New Covenant era the motivation is internal, it is the work of the Spirit in the life of the believer. The internal motivation is also called the “law in the heart” (Hebrews 8:10-11).