How New Testament is relevant when compared to Old Testament.

The Law and Grace:

Law cannot save, Only God's Grace saves.

Matthew 5:21-30English Standard Version (ESV)

Anger

21 “You have heard that it was said to those of old, ‘You shall not murder; and whoever murders will be liable to judgment.’ 22 But I say to you that everyone who is angry with his brother[a] will be liable to judgment; whoever insults[b] his brother will be liable to the council; and whoever says, ‘You fool!’ will be liable to the hell[c] of fire. 23 So if you are offering your gift at the altar and there remember that your brother has something against you, 24 leave your gift there before the altar and go. First be reconciled to your brother, and then come and offer your gift. 25 Come to terms quickly with your accuser while you are going with him to court, lest your accuser hand you over to the judge, and the judge to the guard, and you be put in prison. 26 Truly, I say to you, you will never get out until you have paid the last penny.[d]

Lust

27 “You have heard that it was said, ‘You shall not commit adultery.’28 But I say to you that everyone who looks at a woman with lustful intent has already committed adultery with her in his heart. 29 If your right eye causes you to sin, tear it out and throw it away. For it is better that you lose one of your members than that your whole body be thrown into hell. 30 And if your right hand causes you to sin, cut it off and throw it away. For it is better that you lose one of your members than that your whole body go into hell.

Jesus did not bring new or different law, but a renewal to the Laws - Its the fulfillment of the Laws and prophecies. His comment, commission is the reflection of the Old Testament Laws. It is the "Heart of the Matter" that matters. What causes to sin is just a thought. If we were to Love God with all heart, mind, soul and strength. God is merciful enough to bless us.

1 Corinthians 2:9 ►
Verse (Click for Chapter)
New International Version

However, as it is written: "What no eye has seen, what no ear has heard, and what no human mind has conceived" -- the things God has prepared for those who love him--

Hebrews 10:4-6New King James Version (NKJV)

4 For it is not possible that the blood of bulls and goats could take away sins.

Christ’s Death Fulfills God’s Will

5 Therefore, when He came into the world, He said:

“Sacrifice and offering You did not desire,
But a body You have prepared for Me.
6 In burnt offerings and sacrifices for sin
You had no pleasure.

Not that the Old Testament is completely irrelevant, but it paved a way for the Savior to help us from our sinful nature.