The V-Spec adds a number of changes over the base GT-R, including ATTESSA-ETS Pro (reg. GT-R is not "Pro", not sure what is actually different between the 2 systems), Race ABS (still lets you slide if you want), improved suspension components (lower and stiffer shocks/springs, larger sway bars, not sure what all else, possibly more chassis bracing?), different badging, clear vs. amber signals and markers, more features in the in-car HUD, including datalogging, functional full underbody ground-effects, slight interior changes, different color options, and a few other things I can't remember right now.
The M-Spec is nearly the same as the V-Spec, yet with a more luxurious, leather interior with more comfort options, a softer, more comfortable suspension, more sound deadening, and a few unique trim and color options. While still a GT-R and more capable than most other cars it may encounter, the M-Spec tends to focus a little more on the "GT" and less on the "R".
To my knowledge, the only change made in the V-Spec II from the original V-Spec, aside from badging, is the addition of a painted carbon fiber hood with a functional, raw-carbon NACA duct over the turbos/manifolds.
The Nur models were made to commemorate the R34 breaking the 8 minute barrier on the Nordschleife (which I was more than capable of doing in the game w/ my stock V-specII Nur), and feature a number of minor trim, badging, color, and interior touches to add an elegant air of exclusivity. The in-dash HUD adds a lap timer, and I believe that a Nismo shift knob was standard, though I cannot verify that for sure. The most important difference, however, is the N1 engine and turbos, which are significantly stronger and better than the production versions. I am not aware of any differences or improvements made to the N1 engines, though I am fairly confident they are the same. As far as the Nur models making more power in game than the N1, I have no idea why this would be, though none of the power modifications in the game appear to the most realistic, so I would assume this is just a mere anomaly and implies no differences to the real cars.
Here's some more info on the Nur models specifically:
http://chrisnz.com/nissan-r34-gtr-mspec-nur/