안 vs 못

The great divider.

답장을 안 보내서 미안해요
답장을 못 보내서 미안해요

How would you literally define these two sentences?

"Since I didn't send you a response, I'm sorry."
"Since I couldn't send you a response, I'm sorry."

But, the former is most natural in English, right? "Sorry I didn't respond to you sooner" is the feeling you want to express. However, it may not come off like that in Korean. Dare I say it would make matters worse if you chose the former instead if the latter.

안 means "not" as in 안 했어 I didn't (숙제 안 했어 I didn't do my homework)
못 means "can't" as in 못 했어 I couldn't (중국어 목 해요 I can't speak Chinese)

But, in Korean, these don't translate exactly over 1:1. They don't always mean that.

안 carries the feeling that you had the opportunity and yet you chose not to do so. 못, on the other hand, feels like the choice was not up to you and that outside powers prevented you from doing whatever it was you were supposed to do.

문자 못 보내서 미안 would be much more appropriate to a significant other than saying
문자 안 보내서 미안 would would indicate that yes, you could have sent her a text message but you chose not to. what of it?

Culturally speaking, it's a bit different. In English, unless someone holds a gun to your head, no one makes you not do something. So, if you didn't send her a text, you could say with solemn regard that although you chose not to, you are sorry. No one made you not send it, right? You were having fun with your friends and you lost track of time. No biggie right?

In Korean, it is a whole lot sweeter to say that you were unable to send it. It carries that same feeling that you are indeed sorry for not touching base but think not of it like "I was physically unable because the phone was broke" instead, think of it as "I totally lost track of time" or "정신 없어요". So, although it doesn't mean the exact same in English, the difference is enough to point out.

Another example that is even more curious is 아직 못 봤어요. This seemingly means "I was not able to see it yet" but it really means "I haven't seen it yet." The situation: a coworker asks you if you have seen the new summer blockbuster movie. You casually reply that you haven't seen it yet. Of course you've had time to see it. It's been out for a week or so. You have time after work but you chose not to go see the film. No harm in that. However, replying "아직 안 봤어요" while not grammatically wrong, is not as common as saying "아작 못 봤어요". The idea is that you do want to see it but you haven't had the chance yet. You'll see it soon enough, though.

In summation, Koreans lie have a different way of expressing this type of feeling.