I recommend West to East because once you start the main part of the climb, the majority of the turnoffs will be on your right, and you do not have to turn across traffic. There are not that many places to pull over on the west side of Logan Pass, but the ones that do exist are on the right (once you pass the horseshoe bend). If you have never been there before, you will want to stop and look at the views/take photos/be awed. If you have an issue with heights and no or low guard rails, choose the east to west direction, which will put you on the mountain side of the road in the most exposed portions.
Don't expect to be going faster than the posted speed limits on the west side of the Going to the Sun road due to traffic. The east side is not as OMG spectacular, but still pretty damn good. If you want to ride fast, do it outside of national parks - there are too many people, too many vehicles, and too much wildlife in the park to go much over the low speed limits.
The other commenters are correct about road construction and traffic delays - look at the Park Service website for info. 2010 is the centennial of GNP, by the way.
Timing is an issue for riding the Going to the Sun road. There is no way to accurately predict when the road will be cleared of snow and open the whole way, other than sometime in June. Do not plan on riding it in May - it is highly unlikely that the whole road will be open by the end of May. I've seen snow in July and August on Logan Pass (the top of the Going to the Sun road) - I've even skied on Logan Pass on fresh snow in August. This year, I had rain in early July (July 6 and 13) in Glacier and in St. Mary, and spitting snow [nothing serious, however] on Labor Day just outside St. Mary. I've also ridden in September in other years when it was beautiful. Mid-June to sometime in September is the window of the pass probably being open; July and August are the safest bets for it being passable.
If you are going to be on a bicycle, note that the shorter climb is east to west, and that bicycles are prohibited on the west side of the pass from 11:00 am to 3 or 4 pm, due to traffic.
As noted by others, Waterton Park in Canada is very scenic, but you will need a passport to cross the border. The Beartooth Highway, from Red Lodge, Mont, to Cooke City, Mont. (via Wyoming) is also spectacular.
It is not mere accident of birth that I live half way between Glacier and Yellowstone parks.