This question comes up a lot, understandably so. Some comments from 2010 http://www.350z-uk.com/topic/33288-is-the-370z-an-improvement-over-the-350z/page__st__80
The 370 is an evolution of the 350 so don't expect a huge jump forward. It is more refined, the interior is much improved, sound system is great and I love the sat nav. The best feature is the syncro rev matching though, it makes you feel like a professional rally driver.
I moved up from an early 2005 350z so jumping from 276bhp to nearly 350 after a few mods felt like a big jump to me, the car pulls far more all the way to the line. However others who had later 350s did not notice much difference. I remember Wasso and Andy commenting hat if they didn't look down into the cockpit they would think they were back in their 350s.
The gearbox is still quite agricultural, no change there. The stock 370 sounds terrible, it was a huge downer for me, the car needs a custom exhaust. I also find the 370 tramlines far more than my 350 ever did. A number of people have also commented that there is far more road and tyre noise in the 370. Also found there to be more room in the 350 and the seats were more comfortable, I'm 6ft 7 So find the cockpit a little tight
I do miss the sound of my old 350 and I felt the 350 was a more raw driving experience, you are certainly aware of the technology in the 370 and th driving experience is lessened as a result. In essence the 350 has grown up, it's a nicer place to be and I love the looks more (especially in roadster format) but t growing up it has lost some of its charm but gained other characteristics I like.
Which would I have, I'd take my 370 over the 350 any day but I still love and miss my old 350.
Now as to a 370 vs a Cayman, different question altogether. The Cayman is a superior drivers car in pretty much every way, it's a fine tuned scalpel compared to a pretty brutish gt car. Zed has more power by the Cayman would run rings around it. it all depends on what you fancy, take a test drive in both. Personally I see too many porsches around and wanted something more unusual and I'm happy with that for now.