Danforth allows Elizabeth to speak to John because he hopes that she will convince him to confess to witchcraft. He says to her, "Be there no wifely tenderness within you? He will die with the sunrise [....]. What say you? Will you contend with him?" There is talk in the village of rebellion, as there was in a neighboring town, and, as Reverend Parris points out, the people scheduled to hang today are well-respected, not like many who have been hanged before. If they can get one person to confess, it will seem to validate the convictions of all the others condemned to die.
Elizabeth has changed toward her husband in that she is now willing to accept some responsibility for the problems in their marriage. He had the affair, yes, but, she says, "I have sins of my own to count. It needs a cold wife to prompt lechery." She says that she was always suspicious of his love because she felt that she did not deserve it. She no longer blames him alone for their problems; she blames herself now too.
Elizabeth doesn't take Hale's advice because she feels that she cannot instruct or judge him. She says, "Do what you will. But let none be your judge. There be no higher judge under Heaven than Proctor is!" She believes that John is an incredibly good man, despite his own self-doubt, and she wants him to make the decision about what to do for himself. In the end, she is right; he chooses to keep his integrity and die, redeeming himself and seeing his own goodness once again.
Danforth allows Elizabeth to speak to John in order to get him to enter a guilty plea and save himself. Danforth likely knows Proctor is innocent. However, for the integrity of the court, he cannot simply overturn the guilty verdict. So he needs John to confess in order to appease the growing discontent of the townspeople toward the court.
Elizabeth realizes that she was cold and distant toward her husband and helped drive him away, which is one reason for his affair with Abigail.
Ultimately, Proctor tears up his confession because he wants to keep a shred of dignity. Signing a false confession, in his view, would be worse than dying for a crime he didn't commit. He chooses to die at the hands of an unjust court rather than owe the rest of his life to a lie.
Elizabeth doesn't take Hale's advice because she has recognized that her husband is a strong, proud man who values his reputation and his independence. She doesn't want to be responsible for helping to take that away from him. She has to let him make his own decision.