I really want to know which one of them better for running Unity 3D or GNOME 3 Shell. I'm specifically interested in this for desktop, not laptop, computers.
I recommend them in the following order according to problems found, how the "out of the box" experience feels to end users, how compatible it is and how it will feel once you have it set up:
Intel - Works out of the box PERFECT. Most if not all will just work with no problems. Great for using a Video Beam or another video output. Just plug and done. Feels good also. Of course the benefit relates to how fast the Intel card is. I would not recommend an Intel old 440 to a more recent Core 2 Duo and up versions. In general you should not have any problems with them.
NVIDIA - Great Performance with Unity. It works out of the box for most Nvidia cards by using Nouveau and if you install the easy to install proprietary drivers it will feel perfect. Has little problems that can be quickly solved if you happen to have some of the cards that have small details (Which can also be solved if you update 12.04 with the latest updates). Most cards work excellent with Unity. I have tested cards starting from a Geforce 4600 TI 128MB up to a 560 TI 1GB.
ATI - Also great performance with Unity. It works out of the box although not most as mentioned with Nvidia. If you install the proprietary drivers you will get a good jump in performance but again, this is not as generally mentioned as Nvidia. My curiosity lies in that ATI is more open to the open source environment than Nvidia and yet has a bit more problems with video.
So in general if I had 3 laptops to choose from and they all were the same except for the video card, I would definitely choose Intel over Nvidia, and Nvidia over Ati.
In the case of 3 Desktop PCs that are the same except for the video card I would choose Nvidia over Intel and Intel over Ati.
This is just from several years of good, neutral and bad experiences.
For cases where you have 2 integrated video cards (Nvidia and Intel or Ati and Intel). The performance benefit is not as important as how well the video cards behave on Low power, high heat situations. For information regarding this I always point to Bumblebee or Ironhide?
In general, Ubuntu will work without problems or at least with a minimal of tweaking. The performance gain in the past months for all 3 video card types and the amount of fixes they have received just lets us know that there will be fewer problems for each new version of Ubuntu.
I think any of them has enough "grunt" for Unity.
The Intel devices probably have the best driver support in Linux, because the Intel drivers are Open Source. If you don't need lots of power for 3D apps and games, Intel graphics are probably sufficient, and even preferable, and certainly more economical as they come built into many motherboards.
Unity should run fine with any of those drivers.
Generally, the proprietary -fglrx and -nvidia have higher levels of OpenGL support than the corresponding open source options, -amd and -nouveau, particularly so for newer hardware. The Unity guys did a lot of the initial development with -nvidia since it had better OpenGL than the others.
-intel is a good driver, but it's OpenGL support has tended to lag that of the other drivers. However this has improved a lot lately and for Unity's purposes should be more than adequate in Precise.
the answer is a little complicated, but goes like this:
nVidia's open-source driver is really bad (thanks to them being stubborn), but their proprietary driver gives pretty much the best performance of any GPU on linux.
AMD's open-source driver is better, but still not really good enough to match your investment in a modern video card, so you'll end up resorting to their proprietary driver, which is not as good as nVidia's.
so: if you're serious about wanting to go entirely open-source, AMD is a slightly better option (though still dissatisfying -- honestly, Intel graphics are probably the only ones whose open-source driver compares well to their Windows performance nowadays, and that wasn't true until 2012), but if you're willing to use proprietary drivers, nVidia is best.
note that this assumes that you're using <5 year old hardware, willing to spend north of $100 on your GPU, and have some interest in modern games.