From the side its looks different. If you compare the 350 and 370 side by side, the 370 has a new roofline from the front pillars back, new door shape, side windows are different.
Looks like they have been listening to us and given the side profile a more S30-ish look. The roof line now seems to extend further to the rear of the car, whereas the 350Z's roofline curved down more after the cabin, giving there rear that toad rear end look.
The window shape is more S30, the Z33 looked more Z32. The line that used to run across the top of the guards, door and all the wat to the back of the car is gone, and now the line drops down across the door and kicks up at the end of the door forming that S30-esque rear quarter window.
If the 370Z in the picture wasn't black the differences would be more obvious.. which gives me an idea: