I've got a pair of both. I find the Nike Air (Pegasus) offer me more heel support than the Asics, which are gel filled. Like most things, it's up to the individual and what's right for one person isn't right for another.
Asics, when i bought proper running shoes after a visit to a video your running joint, the first pair were saucony hurricaine 7. They were ok but a million times better than the Nike piece of crap i was using before with totally collapsed heels.
After a while i went for a repeat visit and bought asics and have had em ever since.
Asics for me, I've tried other brands like NB, Mizuno and Saucony but I've always ended up coming back to Asics, however as others have said it's all down to personal preference. The only way to choose a good pair of trainers is to get out and try on lots of different brands/models and even then you might not get the "perfect" trainer for you on your first couple of pairs*.
*Edit - By this I mean you might end up getting a pair that feel good but find out after you've got some miles out of them that they don't work for you, finding the perfect trainer can take a bit of money depending how willing you are to run in trainers that just aren't quite right.