Another potential soon-to-be-NZ-academic in here. Welcome!
I accepted an offer at Auckland several months ago and the TEU site was invaluable. Made negotiations rather straightforward. Auckland has a central appointments committee that will assess your c.v. and decide at what grade you can come in--not much you can do about it if you don't like it. But the "allowances" are really just extra salary, and there's scope to negotiate a bit there. Other things like relocation packages are the same regardless: one way flights, one 20' container for singles or couples (40' for families with children), 21 nights accommodation in an apartment-style hotel for your first few weeks.
Another consideration: what kind of migrant visa. The uni will officially only help with Work to Residence. I'm not willing to move all the way to NZ on a work visa so I elected to apply for Skilled Migrant Category residency. It took longer (7 versions 2 weeks), but it saves me having to apply all over again a few months or years later. There's an in-house immigration consultant, but I didn't use them: if you do you're giving that consultant the right to share all your info with the university, which I found unacceptable.
What department/faculty/campus?