While Nz is technically made up of the waste of Australia, it has been isolated for 60-80million years which even in geological terms is a decent amount of time.
I think both Australia and Nz have both been moving north in all that time as well, so they have come up from the south. The temperature difference would account for most differences in the animals with crocodiles and snakes needing warmer temperatures to survive.
The spider thing is a little different. NZ does have the native Katapo which is related to the Australian redback but it is rare in NZ, only living on some east coast beaches in driftwood (the only one I have ever seen was at university in the laboratory..and I have lived on the east coast my entire life!)...AND I don't think it has actually ever killed anyone.
We do have several other spiders that have come over from Australia..eg I believe there are a couple of redback communities in the North Island...There is also the white tail spider is quite common, which is not poisonous as such but the bite gets infected easily.
Perhaps the reasons for being poisonous arose after the split from Australia, in that there were more things to kill? NZ also has no native land mammals apart from 2 species of bat, while Australia has quite a few things hopping around (marsupials etc).
While some of these poisonsous things may have been present (or ancestors of) on NZ they may have lost the need to poisonous, not evolved it at all or the speices died out.