They mean the same thing, but 안 can only be used with pure Korean verbs, like 먹다 or 하다. 지 않 can be used with both Korean verbs and Sino-Korean verbs (verbs that originate from Chinese). When in doubt, use 지 않 to play it safe since it is always correct. Note there are a few Sino-Korean verbs that can use 안, such as 통하다, but they are exceptions.
Kungbohada means to Study, but also to Learn in the sense to being in the process, it is used interchangably when used in the present form in English BUT - and this SEEMS to be arbitrary (some comments seem to hint at something to be taught later, but I cannot see any explanation here) - Duolingo Korean insists on using Learn in some cases of Kungbu-X and Study in others... why? this is not something I am able to discern (and in English, at least in present tense the verbs are interchangable so it makes no sense that in Korean there should be a difference, tbqh) and it is an increasingly aggravating experience - very needlessly so. WHAT am I overlooking here?