There are also another two variants- прошу прощения and извиняюсь. 1. извините, простите, прошу прощения are equal regarding politeness. You can regulate it by switching between Plural and Singular (Ой, извините! - ٌWhen you stepped on someone's foot. Ой, извини! - the same but the victim is obviously younger than you). Извиняюсь is less polite, kind of low-class, but still not rude, tends to be used by uneducated persons. 2. Sincerity in извините and простите is the same, it's expressed by the intonation. Прошу прощения is a bit less sincere and more cold and reserved but again it depends more on the intonation. Извиняюсь normally implies that you don't really regret much. I would say извините is used a bit more often than простите though the difference is not that big. Простите is slightly stronger and implies a more serious fault. If the verb is followed by object простите is prefered (прости меня). извините, простите, прошу прощения can be used for drawing someone's attention.

Indeed there is no big difference between two these words in my language, but when you use them just remember - "Извините" - Imperative mood from "извинить"=to excuse, but "Простите" - Imperative mood from "простить"=to forgive. We also use "Извините" when we want to ask something, for example "Извините=Ecuse me can I ask you?". And we use "Простите" if we feel and admit guilt because of something, for example ""Простите"= Forgive me, I forgot that you were waiting for my message". Good luck!

They are synonyms

"Простите" - means more deep stage of repentance than "Извините" "Извините" - is comon, but it according to traditions of region.

They are the same