They're similar enough that most Americans can't tell the difference between the two (which is why so-called "Australian" accents in American movies are really Boston accents), and when I was in the Netherlands, which has some of the most fluent non-native English speakers in the world, I was constantly mistaken for British. When I do a language exchange, people that have never heard an Australian speak before can generally understand me just fine (largely depending on the fluency of the person I'm speaking to) I often get told that I sound British.
Australian accents ARE different from English ones, but I feel like Australian accents are generally much closer to Received Pronunciation than some accents within England. Accents don't vary as much within Australia as they do in other countries: speakers from big cities like Brisbane, Sydney, Melbourne, Darwin and Perth sound pretty much the same, but if you travel to rural areas you'll find that people speak with a much stronger 'twang'.
In school we're taught to use British English grammar and spelling, and the Oxford Dictionary is the preferred dictionary here. There's an official Australian dictionary called the Macquarie Dictionary, but think the only times I've ever used that were in primary school. The grammar and vocabulary is pretty much the same as in England (everything I've ever written on italki is how I would normally speak), with some exceptions. Apart from news and game shows, our TV is overwhelmingly American and we've picked up a lot of American vocabulary, especially slang. There are lots of instances when we consider both the American and British versions of a word to be synonyms without realising that they're regional differences. Like any place, we also have our own slang.