ASUS generally make better boards, but their high end boards are completely poo (although still better than Gigabyte). Stay away from the Striker II series, as 1 in 3 are faulty off the production line (ranging from small faults to failing to post - the famous CPU INIT error you see throughout this forum).
If you want a low-mid range mobo with some good basic features, Gigabyte are ok. ASUS provide some good enthusiast boards with the PHEONIX bios utility, but their boards have poor compatibility with some hardware.
So I guess:
Low-Midrange : Gigabyte - almost any
Highmid-Highend : ASUS - Rampage, Maximus, Striker (not Striker II), P5 Series