Go NZ!
Reasonable travelling distances, fantastic accessible scenery, great beaches, both cove-like and miles of golden sand-like, proper little towns, a UK style B&B culture, variety at every turn, a living indigenous culture.
Note the comments re weather, though! Invercargill in mid winter is like Shetland! When the wind comes from the south there ain't nothin' between you and Antarctica!
Oz: Well, vast distances are the real bugbear! You really do have to fly between the bigger cities. The Barrier Reef IS worth a look-see and is the rainforest but far north Queensland (FNQ as it's known) is either very rugged or very touristy. Melbourne in winter, especially in an older, traditional house can be blardy cold, what with wooden walls and no heating! Sydney is a fun, vibrant city but you could 'do' the sights in 2-3 days. Brisbane really isn't a fun tourist destination, you'd need to go a fair way north to get to where the tourist magnet stuff is.
I've spent months in NZ before both as a penniless backpacker and rather more cashed up. I've enjoyed every trip there and would willingly go back (though perhaps not to live!). I lived in Oz for 15 years, again as a backpacker, when I did The Grand Tour, and as a local, in Brisbane and on the Sunshine Coast, near Noosa.
So my feeling is as a one-off destination, I'd go to NZ and get a hire car. Don't try and do everything- if you have a fortnight, I'd do one island only (north), 3-4 weeks, both in a figure of 8.