I think the best way to explain it is simply say its really a completely different car that just looks similar. A 350Z is a Z33 and a 370z is a Z34 using the Japanese model numbers. Nissan does not change these numbers just because the displacement change. The car was basically completely re-designed from the ground up.
They definitely drive significantly different (I have significant driving experience on both). Here is a better comparison pick showing my dad's 350Z and my 370Z. Both are stock. He has an enthusiast (regrets he didn't go performance) and I have a base w/sports:
