My shoes are Nikes - because they suit my running gait and, just as importantly, are a very wide shoe (this particular model is, anyway - the Structure Triax 11). I have unbelievably wide feet and have great difficulty getting any kind of shoe to fit, so I was amazed and relieved to find that these fit so well that I can actually do the opposite to what everyone else does with running shoes and wear a size smaller than I normally wear!
But of course, that would mean that they would be rubbish on someone who has very narrow feet.
So I suppose what I'm saying is the same as everyone else has said - it all comes down to what works best for you. Everyone has different size and shape feet and what is fantastic on one person is rubbish on another. As long as they are decent quality proper running shoes, whether they fit you well and suit your gait is going to be way more important than a slight difference in the way the sole is put together, or the particular fabric the upper is made from.
Even if Asics are the best quality, technically, if they don't fit your feet, they'll be the worst thing you can buy.