**121 The Old Testament is an indispensable part of Sacred Scripture. Its books are divinely inspired and retain a permanent value,92 for the Old Covenant has never been revoked
1963 According to Christian tradition, the Law is holy, spiritual, and good,14 yet still imperfect. Like a tutor15 it shows what must be done, but does not of itself give the strength, the grace of the Spirit, to fulfill it. Because of sin, which it cannot remove, it remains a law of bondage. According to St. Paul, its special function is to denounce and disclose sin, which constitutes a “law of concupiscence” in the human heart.16 However, the Law remains the first stage on the way to the kingdom. It prepares and disposes the chosen people and each Christian for conversion and faith in the Savior God. It provides a teaching which endures for ever, like the Word of God.
1965 The New Law or the Law of the Gospel is the perfection here on earth of the divine law, natural and revealed. It is the work of Christ and is expressed particularly in the Sermon on the Mount. It is also the work of the Holy Spirit and through him it becomes the interior law of charity: "I will establish a New Covenant with the house of Israel. . . . I will put my laws into their hands, and write them on their hearts, and I will be their God, and they shall be my people."19**
The key to the difference between the old and new covenants is that, while the law is incapable of justifying man, it nonetheless must be fulfilled- and we’ll be judged by it (Rom 2), because it’s holy, spiritual, and good. The problem is that man is not spiritual, having divorced himself from God. The purpose of the Atonement is reconciliation of God and man, so that God can truly become the God of man again, the two communing, His Spirit dwelling within so that man can fulfill the law, but the right way. This is the essence of the New Covenant: "Apart from Me you can do nothing."
So the difference is not between whether or not the law must be fulfilled; the difference is in how it’s to be fulfilled: by the OC way: via man’s own efforts, still apart from God, or by the NC way: via the power of God (grace) as we turn back to Him in authentic faith, moving on to hope and ultimately love, the fulfillment of the law. This is how God puts His laws into our hands and writes them on our hearts.