Australians tend to coopt a lot of (originally) NZ stuff, if it/they get famous eg Phar Lap, Fred Hollows, Crowded House, Russell Crowe... In sports, they are 'our' worst enemies, but otherwise are generally just given a hard time because we actually don't mind them - sibling rivalry and all that. New Zealand is even more at the @rse-end of the world than Australia and it's rather small, so they tend to feel like they're in Australia's shadow most of the time and perhaps they have 'something to prove', or at least they need to differentiate themselves from big, bossy, boorish Australia(ns). Australians tend to think New Zealand is cold, wet and just that little bit dull...
When the states of Australia joined to become 'Australia' there were noises made to get New Zealand included too (a reasonable idea) but the New Zealanders would have none of that. In the past the economy was better in Australia so a lot of New Zealanders moved. Migration has recently been tending the other way. About 350,000 New Zealand citizens (out of 4 million or so) live in Australia and around 60,000 Australians (out of 20 million) live in New Zealand. Political and economic ties are very close.
Their accent is a bit weird, though, and they are said to have a thing for sheep.