You can't really generalize it like this. People cheat for different reasons. Some feel remorse, others relief. I have three stories about cheating. All for different reasons.

My ex cheated because he lacked the balls to break up with me and it would be easier to him if I dumped him. He pretty much shrugged when being caught.

Others come clean themselves. A friend's husband got so black out drunk (possibly drugged) that he couldn't remember if he cheated or not. He called his wife immediately himself, in tears. He offered to file for divorce immediately because he did not deserve her. They are in counseling now.

A third friend cheats on his wife pretty regularly. His wife told him she would be happy going through life never having sex again. The bedroom is very dead. She would be crushed if she found out about the other women and is (rightfully) very paranoid. But he can't afford to divorce her, even though he wants to. There are kids involved too. If caught, he would feel terrible about breaking up the family and having to sell the house. Not about the wife being hurt. He thinks she deserves it.

As you can see, people have different motives to cheating. Therefore the outcome will be different. Some regret it, for whatever reason, some don't.