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.