바빠서 전화를 받지 않았어 = 바빠서 전화를 안 받았어 => I didn't answer the calls because I was busy. (chose not to) 바빠서 전화를 받지 못했어 = 바빠서 전화를 못 받았어 => I couldn't answer the calls because I was busy. (was unable to) 안 (short for 아니) and 않다(short for 아니하다) are two ways of negating a verb or adjective, both widely used. 못 and 못하다 are close parallels of 안 and 않다, with the distinct meaning of "be unable to" or "cannot". - (나는) 그런 음식은 안 먹는다 = 그런 음식은 먹지 않는다: I don't eat that kind of food. - (나는) 그런 음식은 못 먹는다 = 그런 음식은 먹지 못한다: I can't eat that kind of food. - 독일어 공부는 안 해 = 독일어 공부는 하지 않아: I don't study German. - 독일어 공부는 못 해 = 독일어 공부는 (하지) 못해: I am unable to study German. - 시끄러워서 공부를 못 해 = 시끄러워서 공부를 (하지) 못해: I can't study because it's too noisy. - 이 방은 안 넓어 = 이 방은 넓지 않아: This room is not spacious (adjective case - 못 cannot be used with adjectives). There's a tricky grammatical subtlety on 못 between 못하다 and 못 하다. 못 by itself is an independent adverb, while 못하다 is an auxiliary verb which must be written as a unit with no space. So 공부를 못 하다 (adverb 못), 공부를 하지 못하다, and 공부를 못하다(하지 is dropped from 하지 못하다) are all correct. (There is another regular verb 못하다 meaning "be poor at something, inferior", so 공부를 못하다 has two meanings). For all non-하다 verbs, however, it is always "못_verb", as in 못 가다, 못 먹다, with a space. There's also -지 말다. It mirrors -지 않다/못하다, but is used in imperative sentences and certain statement contexts that emphasizes intentionally not doing something. - 오지 않으세요: (Someone) is not coming. (않다 negates it - the sentence remains a statement) - 오지 못하세요 = 못 오세요: (Someone) is unable to come. (similar, 못 negates it with "unable") - 오지 마세요: Please do not come. (말다 means acting on one's will, so 마요 and 마세요 are asking not to come). All these details get complex (especially the distinction between 못 하다 and 못하다, the auxiliary and regular verb 못하다), so that even native Koreans write them wrong all the time. So there's no need to feel bad if you don't understand some of it.

short form: 안 + Verb/Adjective, longer form: Verb/Adjective + 지 않다 It is used to negate an action or state. short form: 못 + Verb, longer form: Verb + 지 못하다 It expresses the subject's lack of ability to do something or the fact that something does not go according to one's wish or hope or due to some external factor. It corresponds to the English 'cannot'. 바빠서 전화를 받지 않았어. I didn't received a phone call because I was busy. 바빠서 전화를 받지 못했어. I couldn't receive a phone call because I was busy.

The difference between ~지 못하다 and ~지 않다 is difference between cannot and don't. If you use ~지 못하다, it means You cannot ~. However ~ 않다 means you don't ~. It's the difference. Therefore, if you use 공부 안하다(공부하지 않는다), you say 'I don't study.' On the other hand, if you use 공부하지 못하다, you say 'I cannot study.' In addition, we say 공부못해요 means be slow to learn or to do pour in school.

This is just an over-simplified statement but the basic rule of thumb is that former is used for something under your control whereas the latter is not under your control (forced by situation whereas your will and desire plays no part in).