Both of these mean the same thing and they both give negative connotations to the verb. It's just when they're used. ~지 않습니다 can be added to both action and descriptive verbs, for example: 하다 (to do) ---> 하지 않습니다 (to not do) 먹다 (to eat) --> 먹지 않습니다 (to not eat) An example sentence: [jeoneun bab bbangeuro meogji anhseumnida] 저는 밥 빵으로를 먹지 않습니다. I don't eat rice with bread. (very polite) However, 안 is basically a shortened version of ~지 않습니다, except that it can be added in front of the verb and this version only works on action verbs, not adjectives. For example: 가다 (to go) -----> 안 가다 (to not go) 먹다 (to eat) ----> 안 먹다 (to not eat) But, for example, 화나다 (to be angry) which is a descriptive verb (adjective) cannot be said as 안 화나다, you have to say 화나지 않습니다 or 화나지 않아요. I hope this helped!

부정형[否定形] ①　(동/형―어간[語幹]) + 지 않다 (높은 말: 않습니다) ②　안 + (동/형―원형[原形]) (높은 말: (동/형―어간[語幹]) + ㅂ/습니다) Both share the same negation function. However, when connecting between sentences, ① is more commonly used. === 좋은 하루를 지내세요.／I wish you a good day.

There are two ways in Korean to form negation. ‘안’ and ‘-지 않다’ are both used to make a negation. The easiest way is to just put the adverb ‘안‘ in front of verbs or adjectives. The second way to use the pattern ‘-지 않다‘. This pattern directly follows by verbs or adjectives. So, negation formed by ‘안‘ is called ‘short type negation’, and the Negation by ‘-지 않다‘ is ‘long type negation’. There is no difference in meaning between the form ‘안 + verb/adjective ㅂ니다/습니다’ and the form ‘verb/adjective + –지 않습니다‘. But, there are some restrictions in ‘short type negation’ 1. For 'noun + 하다' verbs use noun + 안 + 하다 영희는 오늘 종일 공부하지 않았지?(O) 영희는 오늘 종일 안 공부했지?(X) 영희는 오늘 종일 공부 안 했지?(O) 2. Some verbs which have their own special negative counterparts(모르다,알다, 있다, 없다) cannot be used with ‘short type negation’ 그는 그 일에 대하여 모르지 않았다.(O) 그는 그 일에 대하여 안 몰랐다.(X) 3. Long syllable verbs(아름답다, 울퉁불퉁하다, 화려하다,사랑스럽다,좀스럽다) cannot be used with ‘short type negation’ 꽃이 아름답지 않다.(O) 꽃이 안 아름답다.(X)