I've not flown either in First, but the consensus seems to be that the main differences are:
- Seat: QF has a suite with a fully flat bed, similar to BA's, while AirNZ has pairs of seats that do not go completely flat (except for on one plane). The NZ seats have loads of space (only 12 seats vs 14 suites on QF) and are old fashioned but apparently more comfortable than they look, but QF are definitely ahead for privacy and sleeping comfort.
- Entertainment: QF has a slightly more advanced in-seat entertainment system, while NZ hands out personal DVD players to augment the existing channels.
- Food / wine / service: by all accounts they are pretty evenly matched, though there have been grumblings recently on the QF board that the F wine selections have got much worse. Both have reputation for good food & wine; certainly my experiences on both in J have been better than on other airlines.
- Lounges: QF is better at LAX, NZ is better at AKL, and IMHO NZ is better at SYD (though I haven't been into the QF First lounge).
Overall QF is generally accepted to have the better product but NZ First also gets good reviews despite the old fashioned seats. It's also one of the last opportunities to fly F on NZ, as from about May next year it is disappearing to make way for a comined business / first product using the Virgin Atlantic 'Upper Class Suites'.
Each have their pros and cons. NZ fly direct to NZ from SFO 3x per week, which would allow you to avoid horrible LAX. You also might want to bear in mind that if you have trans-Tasman sectors (i.e. between Australia and New Zealand) they will only be in J, as neither QF or NZ offer F on trans-Tasman flights.
I envy you - that's a pretty nice decision to have to make, sfoeuroflyer!