TL;DR: I picked Nvidia.
If you would have asked me this about 10 years ago I would have picked ATI. I have never had problems with ATI cards, until AMD came in. At first they were pretty decent, mostly because of what ATI already had in the works came out as "AMD" cards. But soon after they switched from ATI branded graphics cards to AMD branded, I've had nothing but problems.
Drivers crashed at least twice a day, and anything that used OpenGL crashed after a few minutes if I was lucky. Once a driver update even bricked a card (although it was pretty old at the time). The most stable drivers I have ever had for AMD cards were modified versions, that were made in the spare time of a Russian software engineer. (Which kind of tells you something!)
I have recently switched to Nvidia after years of desperately hoping that AMD would fix these problems, and so far my drivers have NEVER crashed. Although I haven't had my Nvidia GTX 970 for long, I can tell that this card will last me quite a while.
I don't agree with how most of Nvidia's software, frameworks, and CUDA is proprietary. I love how AMD/ATI is committed to open standards and software, and I kind of hate myself for "switching sides" because of it. But because of the instability and lack of support for their older* cards, I went with Nvidia.
*Older as in (relatively) how old the gtx 700 series is to the 900 series in respect to the ATI/AMD cards I was using and what was new at the time.
MeatbagWammo
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