The Asus has the more powerful GPU which means it's more suited for playing games. If you want a certain minimum level of game performance then the Asus will last you longer before you feel the need to buy a new laptop with a more powerful GPU.
Asus is a good brand. They have actually been making laptops for quite some time, but for other brand names. I think they started selling laptops under their own brand name about 4 or 5 years ago. The only thing I don't like about Asus laptops is that the battery is integrated inside the chassis which means you need to send it back to Asus if you ever need the battery to be changed. Unless you are willing to partially take your laptop apart to get to the battery so that you can change it yourself. However, ultrabooks have internal batteries so the argument against internal batteries is moot.
Many of Lenovo Haswell generation laptops (or at least the ThinkPad series) have something called "Power Bridge". That means some of their laptops and ultrabooks have two batteries; one internal (relatively difficult to get to) and one removable battery. That basically means you can switch batteries without having to turn off the laptop.