"Wrong" has a strong connotation. I would recommend saving it for problems that involve math or similarly objective things, not a person's ideas or feelings or other subjective things. For example you can be wrong about how to spell a word in English, but you would not tell someone that they are wrong to want to learn English. A mistake is similar but seems less strong. You can say that a person "He made a mistake" and it seems less like an accusation than "He was wrong". People use it frequently in discussing personal things such as "Going out with that girl was a mistake". An error is similar, but usually pertains to something procedural. For that reason it is often used in a context like computer programming. Fault implies that you are talking about culpability or responsibility, as in "It was my fault that there was an error in the computer program". It can also imply a kind of imperfection rather than a complete error. Something that causes a slight problem instead of outright breaking something, as in "There was a fault in the logic of the computer program, so it produced some strange results, but the program seemed to work fine."