I guess when you say ATI Brand, you mean other companies who make ATI based cards. They are all really based on the same design and chipset, so honestly, there shouldn't be a difference besides that some might have passive coolers(lower end ones, of course) and some might have better a warranty service. Otherwise, I seriously doubt you'd see any difference between two cards...
You can't say ATI is always better when paired with Intel(or vise versa) as there's no difference. It all depends on the cards you're comparing.