I had experience with both companies. I've built many rigs in the last years, using Gigabyte, MSI, Asus, DFI, Abit (more than 16 in the last 7 years).
The only brand to give me issues (in different platforms) was Asus. I can name a few issues like:
1) Mobo sudden Death without any kind of overclocking (Nforce2);
2) Cold boot issue (both Via KT333 and Nforce2 chipsets);
3) Overclock only doable by reviewers with golden samples - not for the Joe average;
4) Horrible webpage for support - slow / time out / dead links;
I had also an Asus Videocard dead on arrival. And the replacement had a noisy cooler.
I would rate something like this:
DFI>Gigabyte>Abit>MSI>>>Asus
DFI: reliable, excellent quality, highly o/c but needs lots of tweaking
Gigabyte: reliable, excellent quality, lots of goodies, good o/c but nothing special
Abit: reliable, excellent quality, good o/c
MSI: stable, good value for the money, lots of goodies, regular o/c
Asus: good reviews, normally more expensive than others, not reliable, poor quality, bad support.
There is also Epox, but my experience was a mixed bag (1 good, 1 bad).
My current rig has Gigabyte inside, my previous one was DFI and the next one will be DFI again - just waiting for a Conroe DFI lanparty mobo.