Difference between 過ち and 間違い

Question

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.

「俺はあくまで身勝手に自分の意地を通すためだけに‌あんたと違う道を選んだ」‌
「これが過ちだと理解はしている」‌.‌

I chose a different path solely in order to have my own way.‌
I'm aware that this is the wrong thing to do.

Answer 1

間違い refers to mistakes people make almost every day, but it only refers to ones caused by one's mental functions. Physical mistakes (e.g., while playing sport) are normally called 失敗 or ミス.

過ち is a stiff, literary and serious word. It refers to a big mistake or a terrible decision that can even affect (or affected) your life. It is often, but not always, related to morals. This word is so big that it is even natural to say 私は人生でたった一度だけ過ちを犯した.

In your example both 間違い and 過ち make sense, but the latter sounds more solemn.

See also: What would be the closest native Japanese word to エラー in the context of IT?