I have had a couple of DSG's the VAG 7 and 6 speed ones, the auto in the 370 has pros and cons with it . It is smoother when pulling away than a dsg box but not so smooth going through the gears, in heavy traffic its much better than a dsg imo. Much more reliable than the dsg though with lower repair costs
Its an excellent 'box imo, you get a 7 speed rather than 6 like the manual , the flappy paddles do really transform the car when you use them with a nice blip when you down shift, like the dsg you can override the auto at any time when driving if you want to overtake and don't want to wait for kickdown. Unlike some tiptronic autos the car will not shift up unless you tell it to, it will however shift down to prevent the engine from stalling.
The auto version is ridiculously over geared, the rev limit in 7th with give a theoretical top speed of 240mph +so it makes for very relaxed cruising and pretty good on fuel , there is little or no penalty economy wise like many auto v manual versions
The gearbox can be tweaked a little with an UPREV map which can make a decent amount of difference especially to the responsiveness
After owning a manual 350z I wouldnt have considered the manual 370z , it had to be auto
The GT Edition and I think most 2012 onwards cars also come with a snow mode which helps traction no end when grip is compromised, not just in snow but wet grass or mud can be troublesome
Good video showing how the cars shift under power