How do the teachings of Jesus Christ differ from Moses? Did not YHWH give the Law to Moses?
They are more developed and detailed. For instance, in the Sermon on the Mount, Jesus develops the teachings on the Ten Commandments. Another example can be found here:
John 8 (New American Standard Bible)
The Adulterous Woman
8 But Jesus went to the Mount of Olives. 2 Early in the morning He came again into the temple, and all the people were coming to Him; and He sat down and began to teach them. 3 The scribes and the Pharisees *brought a woman caught in adultery, and having set her in the center of the court, 4 they *said to Him, “Teacher, this woman has been caught in adultery, in the very act. 5 Now in the Law Moses commanded us to stone such women; what then do You say?” 6 They were saying this, testing Him, so that they might have grounds for accusing Him. But Jesus stooped down and with His finger wrote on the ground. 7 But when they persisted in asking Him, He straightened up, and said to them, “He who is without sin among you, let him be the first to throw a stone at her.” 8 Again He stooped down and wrote on the ground. 9 When they heard it, they began to go out one by one, beginning with the older ones, and He was left alone, and the woman, where she was, in the center of the court. 10 Straightening up, Jesus said to her, “Woman, where are they? Did no one condemn you?” 11 She said, “No one, [a]Lord.” And Jesus said, “I do not condemn you, either. Go. From now on sin no more.”]