Agreed with the poster above...1 week is enough time to cover part of 1 island in a decent amount. Doing both will make you feel like you did not get a full experience in either.

'Nature in general' - I would generally say that South Island has the more scenic views. Culture - this can mean many things...do you mean native/Maori culture? Modern-day NZ culture? Something that's a bit of a throwback (e.g. art-deco in Napier)?