What does John Proctor struggle with in The Crucible that no one else knows?
Proctor does not want to publicly admit to his affair with Abigail Williams.
Elizabeth knows about the affair and the town suspects that something happened in the Proctor house, but John Proctor does not actually admit to his shameful affair until he goes to court in an attempt to save his wife.
As a man of pride (and of practicality) living in a small town, Proctor does not want people to know what he has done. The affair represents a conflict from the outset of the play. From the moment Proctor and Abigail talk early in the play, to Proctor's discussion with Elizabeth, then his confession in court, Proctor is struggling to forgive himself and to acquire forgiveness from his wife for his mistake.
This is the thing that he struggles with that no one else knows about, outside of Elizabeth and Abigail.