Both 過ち and 間違い mean mistake but I think that 過ち has the "moral" meaning while 間違い is more a mistake in the context of wrong/right answer (exam, etc ...)
Am I wrong ?
Here is some context: A detective in the police is writing to his superior (whom he admires and likes a lot) a letter about his future defection since he came to the conclusion that by adhering to the procedures, they cannot capture a certain criminal who has managed to evade conviction so far. So he's saying he is on the verge of leaving to track the criminal alone.
I chose a different path solely in order to have my own way.
I'm aware that this is the wrong thing to do.