오스트레일리아 seems to be derived from the English "Australia", while 호주 seems to be etymologically related to Japanese's 濠洲 (Gōshū). I don't know whether it's derived from Japanese, or they just share a common ancestor (Which Korean country names originate from Japanese? only mentions 호주 in a comment)
In which context is each of the two words used? Also, is the relative usage of 호주 versus 오스트레일리아 changing over time?
My initial suspicion was that if there's a non-English derived form and an English derived form, the latter would be used mainly in advertising to look hip and cool, or that newspapers would use hanja of the non-English form, but that doesn't seem to explain why both words exist in this case. I've mainly been taught the word 호주, but the Korean edition of Wikipedia uses 오스트레일리아 as the title of its article on the country and appears to use both forms within the article.