I've worked in both Aus and NZ. Australian trained, have worked in QLD, NSW and Victoria.
Very similar standards of care etc. In fact many of the colleges (ie college of surgeons, college of emergency medicine, anaesthesia, ICU etc) are Australasian, ie you receive the same training and receive the same qualification at the end of it regardless of whether you're in NZ or Aus.
I've worked in Auckland since 2012 and have been at North Shore, Auckland City and Middlemore during that time. Auckland is drowning under the weight of a rapidly expanding population from a healthcare perspective. South Auckland and Middlemore in particular are at breaking point. There is no hospital between Middlemore and Hamilton, yet where are there development upon development being built? In exactly that area. Out West is on a similar scale, Waitakere Hospital is a tiny hospital never meant to service 500,000 people or whatever it does now. It doesn't even have most services you'd expect from a major hospital.
Staff in Auckland Hospitals work hard, but there is a constant sense of never doing enough, the hospitals are always over 100% capacity, hospital beds are manufactured in the short term, but it's all bandaid stuff.
As for me, I'm moving back to Brisbane in December. More resources, more money, more hospitals. If I got sick, sadly I'd rather be there than here. That's not to say that hospital workers in Auckland aren't as good, or as caring, or any other reason, other than these hospitals are buckling under enormous loads and when that happens mistakes are made, things are overlooked, adverse outcomes increase etc etc.
As for Taylor's daughter and the advice to stay in Sydney. I have NFI what her problems are, but it usually makes sense to stay where you are if you're getting top level care, purely from a logistical perspective. If you've got serious issues that will require years of treatment and followup then having the same people treating you from the outset is in theory better for you. You know them, they know you, they've prescribed the treatment, performed the operation(s) etc etc - the importance of which cannot be overstated, depending on the case.
Hope that helps!