Actually, NZ is definitely NOT part of Australia... Two separate land masses; two separate countries. In fact there's a pretty (mostly) fun rivalry between Aussies and Kiwis, so citizens of each might resent the remark NZ is part of AUS.
There's an old joke that says Australia was a place where England sent all its criminals and ne'er-do-wells... And New Zealand is where Australia sent the worst of the lot!
But it's just a joke. I'm a HUGE fan of New Zealand, and plan to go back again in the next couple of years (from US). Not only is the country itself beautiful, the people as a group are simply the nicest of any country I've been too. And it's a sincere nice, not a polite nice, if you know what I mean.
As far as the continent issue, they are considered the same continent, but should not otherwise be grouped together, any more than Ireland should be grouped with Spain, even though they're both European nations.