Having used both brands of video card in Linux over the years, the tl;dr. is that Nvidia has much better support with their closed source drivers on Linux than AMD does. Here's my anecdotal evidence for why I think so.
I've used three computers that came with various kinds of AMD graphics cards, and all of them have given me nothing but problems in Linux. The first one was an ATI Radeon Xpress 200M, built into an old laptop I bought in 2007. This video card appears to have already been obsoleted by AMD at the time I bought the laptop, but that's another story.
The Xpress 200M card was problematic for both Linux and Windows. It only worked reasonably well with Windows XP; and it's entirely not supported by any means in Windows 7 or 8. In Linux, I can only use the open source
radeon driver with it, but that doesn't give me any kind of hardware acceleration. If I install the
fglrx driver (AMD's closed source proprietary one), it makes the system completely unstable, and random kernel panics and freezes become very common.
My second computer with an AMD video card was a Dell Studio XPS desktop. I don't remember the exact model number of this AMD card, but it was somewhere in the mid-range area. I installed the
fglrx driver in Linux, and it worked reasonably well, except every once in a while my screen would completely go black, and then I could bring back parts of my display by "refreshing" them (i.e. moving my mouse around, dragging a window... any time a part of the screen needed to be redrawn by Linux, it would be redrawn and the solid black would go away). My XFCE panels were particularly difficult to get to redraw themselves, though, because they don't refresh very often. I'd have to kill/restart the panels instead.
The reason I replaced this card with a mid-range Nvidia wasn't because of the random blacking-out issue, it was actually the card's pitiful performance in Windows 7. I ordered the desktop with suitably powerful specs (6 GB RAM, 6 core 64-bit AMD CPU), so that I could run emulators for the likes of Sega Saturn and GameCube. For the latter, the frame rate would be pretty slow in parts and I suspected the video card was the bottleneck, so I tried replacing it with an Nvidia card I had from my old desktop. This did indeed speed up the frame rate in the emulators by quite a lot (most games run at full speed most of the time), and of course fixed my blacking-out issues in Linux.
The third time I had to deal with an AMD card was on a work PC. This one has an AMD Radeon HD 7400 Series video card, and it really caused nothing but problems.
First, the open source
radeon drivers in this case were entirely useless. About half of the time when I booted this computer, it was unusable. I'd end up seeing a completely white screen, with maybe 3 pixels worth of stuff happening at the top of the screen (I think it was the bottom of an XFCE panel, with a workspace switcher applet). It's like the screen resolution was completely wrong and/or scaled up to a ridiculous level. Switching to text mode didn't work either... the screen would go black, but there'd be no prompt (presumably, the prompt was WAY outside the screen borders).
The other half of the time, the display would simply be off-centered. The left edge of the display would be about 1/3 of the way across the monitor, and then it would wrap-around on the right so that the right part of the display was on the left 1/3 of the monitor. Attempting to change the screen resolution within XFCE (using both XFCE's built-in tool, or
xrandr directly), would put the monitor into "seizure mode" where it would flicker black and white rapidly.
fglrx drivers fixed most of my problems, except that AMD feels the need to let me know that my video card isn't officially supported. They placed a watermark in the bottom right corner of my screen, that's rendered on top of everything else the display puts out, that has their logo on it and says "Unsupported hardware". And there's no configurable option where you can say "that's fine, just let me try my own luck using this driver anyway". Nope, to get rid of the watermark, you have to hotpatch the driver binary to basically delete the image out of it, and then reboot. There's a shell script on the Internet that does this - just google "fglrx watermark"
In contrast, I have never seen an Nvidia card that gave me any problems in Linux. The binary drivers for Nvidia have always been absolutely perfect. The only issues I'd ever run into were the times when Fedora would get a new kernel update, and the third party group who package the Nvidia driver lagged behind a day or two in getting their update out. This is largely fixed by using
akmod-nvidia instead of
akmod's automatically rebuild themselves when you update your kernel.