Ask An Australian

Here's a short Wikipedia article on the subject.

The difference in the vowels is the main distinguishable trait that people within Australia can pick up on. In Adelaide you are more likely to hear longer, rounded vowels (see the trap-bath split section in that article) whereas in the rest of Australia they are more likely to be abbreviated and nasally. Personally I think people in Adelaide also have a clearer enunciation on average compared to the rest of the country, though non-Adelaideans don't seem to notice/comment on this difference as often.

What llandar said, but basically we sound a bit more 'posh'.

I've heard it attributed to the fact that Adelaide didn't have convicts, just free settlers. So more refined British accents influenced the way people spoke as opposed to broader and more diverse accents. Could be complete rubbish, not sure.