Aussie here! I know we've got a few New Zealanders on the board, must point them in the direction of this thread so they can offer a trans-Tasman view of how we differ.
We use different money- $NZD and $AUD are listed separately and (in general) the AUD has been stronger against the $USD for the better part of the decade.
Really, it's like comparing Canada to the States - same language, similar geography, similar cultures, a lot of similarities, but don't ever accuse one of being the other!
In my experience, Maori culture (Indigenous NZ population) is a lot stronger and more prevalent than the Australian Aborigines culture has been to modern Australia - when white settlers first arrived here in Aus they pretty much said 'Hey, go away, we want to use this land for a prison and to raise sheep and cattle.' and pretty much subjugated the Aborigines to white rule from day dot, whereas in NZ the Maoris were seen as more 'civilised' and there was (and still is, I think) a treaty signed between white settlers and the Maori's that meant their culture is a lot more preserved.
NZ is, obviously, physically much smaller and a little further south than Australia, it's a lot more mountainous and gets more snow, and is sitting on an earthquake-prone part of the planet. I can't actually recall an earthquake in Australia causing any major damage.
As far as international relations go, in a lot of ways the two countries are like siblings, who argue over a lot of things but really want the same outcome in the end. We're both part of the Commonwealth, both countries are nuts about sport (although NZ is more into the extreme stuff like Zorb and bungee jumping) and we both speak English, have right-hand drive cars and use the metric system.
And I seem to have just muddied the waters rather than making things clearer...