If I had to pick some differences, 엄청 could be used standalone(?) like "와 이거 엄청나!" meaning "Wow, this is great!". You will never hear someone say "아주나". And here's one more thing. 아주 can be used to express the vibes that something has been completed. You can say "우리 아주 갈라섰어", which means "We totally broke up".

아주 is an all-aound word for "very" or "very much". You usually can't go wrong with it. 엄청 (from 엄청나다 = stunningly great in quantity, magnitude, or impact) is more colloquial and has the feel of "shockingly/ridiculously" much. Its use is limited to things that can be easily measured or felt, and would sound funny in more refined contexts. For exameple, - 비가 아주 많이 왔다. (ok) - 비가 엄청 많이 왔다. (more effect in describing an unusually large amount) - 길가에 아주 예쁘게 생긴 여자아이가 앉아 있다. (natural). - 길가에 엄청 예쁘게 생긴 여자아이가 앉아 있다. (unnatural). - 바이올린 연주가 아주 섬세하시네요. (natural - "very refined"). - 바이올린 연주가 엄청 섬세하시네요. (unnatural - "stunningly refined").

we do not distinguish adverbs of degree strictly. 엄청 매우 굉장히. . ..- use whatever is most convenient for you