While the legal code of Moses has been exchanged for the person of Jesus Christ, the law of God itself has never been utterly repealed. It has rather been placed into a new framework so that there has been “a change in the law” (Heb. 7:12, ESV). The law has come under new administration. So it’s fair to say that while we haven’t been given a new law, we have been given a new covenant.
While the Old Covenant presented God’s law on tablets of stone, under the New Covenant, God’s law is written on our hearts (2 Cor. 3:7; Heb. 8:10). So we now serve God in the new way of the Spirit and not in the old way of the written code (Rom. 7:6). While the old way brings condemnation, the new way brings righteousness (2 Cor. 3:9).
While we are not under a condemning legal code, we are still accountable to God’s law. So Christ could say, in Matthew 5:17, that he had not come to abolish the law and the prophets, but to fulfill them. There is a huge difference between abolishing something and fulfilling it. Christ fulfills the law and prophets, not by abolishing them, but by granting a clearer vision into the heart of God than what they could provide.
Both the Mosaic legislation and the pattern of Christ’s life and teaching had some common goals—most notably to reveal the desires of God’s heart. While Mosaic legislation had offered an incomplete picture, Christ gives a clear window into the heart of God. The difference between Mosaic revelation and that of Christ is like the difference between analogue TV and HD. Both provide a view, but one is much sharper than the other.
So why do we still need the Hebrew Scriptures, commonly referred to as the Old Testament? If we’re not under the Law of Moses, why should we study it? Well, how can we understand Jesus, who fulfills the law and the prophets, unless we know what he fulfills? While our aim is not obedience to all the specifics of Moses’ legal code, we need to know what the details of that code reveal about the heart of God. Why did he tell farmers to leave the edges of their fields to be gleaned by the poor? Why did he command the brother of a deceased, childless man, to marry the man’s widow? Could these laws have something to do with care for one’s neighbor?
The Hebrew Scriptures are vitally important to our spiritual well-being. They were the Bible of the early church, the Scriptures Timothy had known since childhood, which could make him wise for salvation (2 Tim. 3:15). We understand more about God’s character and his desire for us by immersing ourselves in this literature.
I would recommend Jay Guin's two blog posts on "God's Plan: Why God Gave the Law of Moses," Part One and Part Two, installments of Guin's multi-part review of John Walton's book Covenant: God's Purpose, God's Plan. The concepts from this book helped me to better articulate how Jesus fulfills the Law of Moses by being God's fuller revelation of himself.