as is the usual, it is best to aim for the middle most part of the season, so to be sure that enuff snow has fallen to form a good deep base, but also so that it hasn't yet started to melt.
the advice aboot temps is a good one, coz whistler being coastal, has far wetter and warmer weather conditions, but for a summer time aussie, anywhere will be bloody nippy!
the main thing i'd suggest you consider are the differences between the places you listed.
big white offers in resort lodgings and limited apres ski, but has very little challenging terrain and is full on 'family friendly'.
whistler is vast, consumeristic to the point of obscene and will be over run what with the olympics and all. but you can live in a village that accesses the piste by gondola.
banff is a village with no actual ski hills, but lots of shops, hotels, restaurants etc. you will have to take a long bus ride to either lake louise, or sunshine village. (well 30 - 40 minutes bus rides aren't long in themselves, but compared to 'ski in' big white or 'home run' whistler they are!)
all have the nice and easy slopes for kiddies and noobs. whistler has the user-friendly side of things. banff can be a base point for some epic technical stuff.
just so you know
Just coz you don't understand it
Doesn't mean it makes no sense!