For the most of their history, Nvidia has made more powerful cards and supported them better by offering more incentives to game developers. However ATi had massive graphical advantages around 2004 and are picking up steam once again.
For the cards that I've had the Nvidia cards tend to have less driver issues. But I do have issues with some of their business practices, that just seem to be an unfair advantage and give incentives for game developers to punish consumers for having an ATi card. Such things as the PhysX card, not giving anti-aliasing options when an Nvidia card is not detected and so forth turn me off.
When I buy something I look at the product and less so the name. However if I had to choose a company, in the end I would side with ATi on a moral standpoint. It's still a close call though, and their competition makes games better.