侘び(わび) The most formal, official word. More than 申し訳ない。 深くお詫びいたします!
(あやまる) To make a mistake (noun form 誤り). Basically this is a more formal version of 間違える! Used by older folks instead of 間違える, but you can use it if you’re talking about a miscalculation or if you took the wrong course of action/ a failed policy.
(あやまる) A casual way to apologize. Often used in anecdotes about a third party.   $$$
(しゃざい) A newspaper word for apology: “The politician gave an apology at a press conference Friday.”