I found that when I was in America, people commented on my accent being distinctly different. This is quite obvious, but the differences between the two broad classes actually have to do with construction of vowels and more specifically the glottal stop. This in fact somewhat strange as both are more or less decended from Irish variants - in the case of America this was because of emmigration and Australia it was more to do with transportation, so the formaer rather than then latter should have a "richer class" of speaker.
Dialect does have a great deal to contribute as well, with Australia picking up Cockney and Northern turns of phrase, reflecting the criminal element on which the country was started. At any rate the techincal differences are vast, and the following are the distinct causes I think.
In many areas the American "t", when not the initial consonant in a word, is pronounced closer to a "d", and in some cases can disappear altogether. Thus latter and butter sounds more like ladder and budder, and words like twenty and dentist can sound like twenny and Dennis. Why do Americans pronounce t as d? Perhaps because to pronounce the frequent "r"s at the end of words ending in "-er" it is easier to say "-der" than "-ter".
In Britain, "t" is generally pronounced like a "t", but there are areas the glottal stop is very well known. This is the sound in between the two vowels in uh-oh, or the initial consonant in honest. In these two examples, and others like them, the glottal stop occurs as much in America as in Britain. But the glottal stop that replaces the "t" in the Cockney and Glasgow dialects is much stronger; imagine bracing for a punch in the belly when you make the sound.
As an interesting side note, Americans sometimes replace the "d" in a British word with a "t", as if hypercorrecting "d" back into the more "correct" "t". I"ve heard "Wimbleton" on ESPN and FOX Sports, found that spelling in the Webster's which is the major American encyclopedia, and whilst looking, even found cases of "Wimpleton". This confusion is borne out by Americans trying to imitate a Cockney accent by putting a glottal stop in place of "d" instead of "t" , which sounds quite odd to an English person.
In Britain, the glottal stop occurs in informal speech in many areas, although with Estuary English, perhaps not informal anymore. The association of the glottal stop with lower classes or Cockneys typically also includes dropping of "h"s , and dropping the g in -ing words.
The other major difference that I was told about specifically with me was that I'll insert "l" and remove "r" from certain words. This lies in the almost unique aspect of Australian English that we have very sharp vowels but are a non-rhotic nation.
Rhotic speakers will pronounce the r in barn, park, cart, fart, whereas non-rhotic speakers won't, making no distinction between barn and (auto)bahn. Most of America is rhotic, with the notable exception of the Boston area and New York City. SE Britain is apparently the source of non-rhotic. England is non-rhotic, apart from the SW and some ever-diminishing northern areas. Scotland and Ireland are rhotic. In the movie The Princess Bride, the bishop (Peter Cook) over-emphasized the non-rhotic accent by loudly announcing "mawidge" (marriage), and Americans often joke about eastern New Englanders who "pahk the cah in Hahvahd yahd".
In Britain, the non-rhotic accent gives rise to linking "r"s, where an otherwise unpronounced "r", in "clear", is pronounced if followed by a vowel, "clear away". An intrusive "r" is an "r" added in such a situation where none actually exists, so "law and order" becomes "law ran order". In some cases, there is even hypercorrection, such as adding an "r" (Louisa - Louiser), especially when a non-rhotic person moves to a rhotic area. But if Clair hears the "r" she'll correct you.
In contrast, in the North and Scotland, r's roll stronger. Even d's can be r'd. I've been called a bluhreeiree (bloody idiot) a few times. Worse is the fact that Scottish and Geordie have stolen words from Gaelic; Scottish Gaelic is different to Irish Gaelic which futher fuels the confusion. The quaint term to refer to one's children as bairns falls hopelessly into incredulity if you happened to mention the great Scotsman Robbie Burns. Inadvertanly Scots and Geordies may actually be speaking about his children, and if "bairns" are in the "burns unit" in "burnside" you'll have a right old mess.
So then, where does this leave me? Well thankfully thanks to McTelevision and the great and powerful Beeb, Australians in general have no problem in understanding Americans or Brits. Conversely because of the way that Australian sounds are constructed, they can in fact pick up everyone else's accents easily with either coaching or by immersion. Americans on the other hand always sound odd when affecting any British accent, except for Renee Zellweger in Bridget Jones, but that's due to very strict & formal coaching.