As far as writing goes, the grammar and spelling are almost exact. However, the main differences are in the accent and adopted words that Australians use but the British never do.
Do not mistake one for the other! ;)
Australia has different influences on its English, mainly from Aboriginal and Mediterranean sources, and occasionally Asian. Eg. "malaka", "cooee", "galah" and "yakka". It's not unusual to describe something as "grouse" or "deadly", and sometimes we even pluralise "you" in to "youse". "Mate" seems to be a floating particle we pepper everywhere. Also, sometimes we shuffle adverbs, eg. "I just got here"->"I got here, just".
We commonly preface a long sentence by saying "yeah, no,".
We seem to throw around swear words a lot more, but we do it without the provocation or anger that usually accompanies bad language. It's almost part of regular speech. However, please do not swear at an Australian - that is provocation. Tricky, huh?
That said, Aussies and Brits understand each other quite well as we share a love of colourful wordplay, and even much of Cockney slang is understood by Australians. Although some of our analogies are far-stretched, eg. "flat out like a lizard drinking"(=busy).
There's a strong Irish influence on the accent, as well as a pile of influences from British dialects. We place our sound differently, a little deeper down towards the chest. It's a warm and easy sound.
Final R's are dropped in speech, and many words are made diminutive: afternoon=arvo/arvie, tradesman=tradie, (swimming)costume=cossie, etc etc
I'm keen to hear what the other Aussies here have to add (or correct). :)