No actually there is a very thin like between the two, difficult to explain still i try:
Fault: - Fault is a mistake but normally used for taking responsibility or blaming someone. like it was my fault or it was ur fault.
Mistake: mistake is normally just conveying that it was ur mistake not necessarily u r blaming that person for his mistake.
Error : error is normally alphbetical or numberical or something in which some typing, calculation etc is involved.
wrong: wrong is not synonemous with mistake or error eg. i do some act which is wrong like murder someone but that may not be mistake or error i may do that wrong intentionally and knowingly however mistake and error are normally unintentionally