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.
I also read on another question's thread that the second form is less formal
For negation -지 않다 are a pair. 가지 않아요. I don't go. 사지 않아요. I don't buy.
So why is not 공부지 being used here? Why using 공부하지 instead? (I don't even know if 공부지 is correct, I'm just asking 'cause I though in this case you just needed to add -지 않다 to the verb stem which I understand is 공부 in this case)
공부 alone is a noun. 공부하다 is the infinitive, thus the verb stem is 공부하-
I tried "I am not studying" and it was marked wrong. Does Korean have a consistent means of marking the progressive versus the simple/stative?
Yes, the progressive form is expressed through the form -고 있다 but they should go over that later.
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?
Why is there a 'ㅎ' below ' 않' ? It's pronounced 'an-sib-nnida' . What's wrong with just' 안습니다'?
공부 means "study (noun)", 공부하 means "study (verb)".
하다 means "to do", and when you put it together with a noun it becomes a verb.