I have not skied in Aussie, but have visited their mountains in summer for hiking. I have not been to Europe so can't compare there.
Australia have tree skiing, New Zealand don't. Skiing in NZ is all above tree line, big open bowls and faces.
In NZ you have to drive us access roads to get to the mountains, little or no on mountain accomodation not enough flat are, and not allowed for environmental reasons). Usually a 30 minute drive form a nearby town. In aussie you can experience the mountain village at places like Thredbo.
The New Zealand resorts apparently get better snow - I base this on the number of aussies who come to NZ to ski instead of staying at home - the snow is better and it is cheaper for them.
If you get to NZ, I would skip the North island ski areas and come straight to the south island.
Web site www.snow.co.nz
has info on the 22 ski fields in the south island, trail maps etc. Many of these are small club fields with rope tows, but are open to the public. If you like staying on the mountain and going back country then the club field are excellent and cheap for this.
There is no worries with altitude sickness, nothing is high enough in either Aus or NZ. because we are so far south it is still plenty cold enough for snow.
You can ski out of bounds in NZ and you will be rescued by ski patrol or police serach and rescue. Unless you were totally negligent they won't charge you.
I cannot compare the NZ fields to any of the Canadian ones I have been too because they all have trees and we don't have any. Don't miss Treble Cone, NZ, if you make it down under. Has the most vertical out of all the fields and is our favourite for variety of terrain.