Sounds like New Zealand is the place for you. With 3 weeks you can rent a camper van and travel around the South Island at your own pace. I am currently planning a trip to NZ for 2015 and it seems camper travels is very popular and you can rent them for $1200 -$1600 USD a week.

In HI hotels are more the norm. I have visited 4 Islands in Hawaii. I especially enjoy Kauai but it's a pretty small Island so for 3 weeks in HI you would be best visiting several Islands. It's fairly easy to get to HI for a week or 10 days but it's much harder to get 3 weeks to go off to NZ.