Well first of all, New Zealand and Australia were never together enough to be "seperated". Prior to colonization the indigenous peoples of both landmass had little to no contact with one another and both countries have since had fairly loose political and imperial connections given their proximity. Australia was a penal colony - essentially a vast arid wastebasket the British used to dump off the dregs of society, but it was an enormous enterprise (if that's the word to use). NZ held very little value to most major nations, and though there were plans to settle it as a penal colony the presence of the Maori people and their fierce warrior culture (quite overwhelming different to the more passive and peaceful Aboriginies of Australia) proved too problematic to be worth the effort. NZ was only ever a pawn in the colonial game and that's why it came under the blanket of british sovereignty.^1

Not to mention that once both countries became settler societies the emergent middle-class of New Zealand and their government were very eager to distance themselves from the black smudge of the convict stain - remember this was in an era where to be a convict was to have something inherently wrong with your genetic makeup, to have a mark that could never be washed off regardless of how many years you were sentenced to in the colonies. This is to the extent that some claim the early NZ government made an effort to conceal and outright deny any connection to Australia's convict past, i.e those convict and ex-convict men and women who came to New Zealand to live amongst natives as Pakeha Maori, those who made up a good deal of pre-treaty New Zealand's European population.^2

This may also contribute to one facet of the kind of relationship the two countries are said to have in modern times - Kiwis often refer to Aussies as "bogans" and "backwards", though almost always in a joking manner. .

1. Keith Sinclair, A History of New Zealand, 1959.
2. Matthew Wright, Convicts: New Zealand's Hidden Past, 2012.