Having stayed at both, the posts on here do not lie, the GH is hands down the better place. For one, it is not that old of a place so everything is still nice and new. The rooms are extemely nice with a huge bathroom that has a seperate shower and tub. The Westin has been around for awhile and is starting to show it. The inside of the Westin rooms look like an early 80s hotel with a heavenly bed.

I did not use the workout facilities at the Westin, but the one at the GH is very nice, with lots of cardio, headsets, etc.

The GH is the best hotel in Seattle in my opinion. Hope this helps.