Proctor is uncomfortable with Abigail's confiding in him on the nights events. He and Abigail had engaged in an illicit affair, and they had been caught by Elizabeth, his wife. This is why she does not trust her husband. Their marriage is in a constant state of turmoil because of it. Proctor does not go to tell what he knows because he feels that he cannot prove that Abigail said it if she decides to deny it. He also does not put a lot of stock into the powers that be.
Furthermore, he does not want to have to discuss why Abigail would be confiding in him in the first place. He is torn up with guilt, and does not wish to cause Elizabeth any more pain or embarrassment.
Elizabeth wants him to go and report what he knows because it is the right thing to do. Elizabeth is a god-fearing woman who believes in telling the truth. The situation is getting completely out of hand, and she feels that this would help.
In act two, Elizabeth asks John to travel to Salem to testify about what Abigail privately disclosed to him concerning the rumors of witchcraft. Proctor hesitates, then says that he wonders if he can prove what Abigail told him since they were in a room alone at the time. Proctor knows that he has no witnesses and cannot prove that Abigail told him the girls were simply caught dancing in the woods. Proctor also doubts that the court will take his word over Abigail's, and he does not trust the court officials. Elizabeth wishes that John would travel to Salem because he has information that can save the lives of those falsely accused of witchcraft. However, Elizabeth has her doubts about John and Abigail's private meeting because she is aware that they have already had an affair. Elizabeth has not forgotten her husband's infidelity and does not fully trust him in the presence of Abigail.
He is afraid that if he reveals Abigail to be a fraud, she will confess their affair. Elizabeth likely want to go with him because she doesn't trust him. Proctor, after all, lied to her when he saw Abigail with a group of people when, in fact, they were alone. Elizabeth mistrusts him because she is aware of their affair.