hi there. I'm from canada and have driven across the country. I'd definitely recommend the west coast - both banff and jasper are fantastic. Highway 3 in the south of BC is a beautiful drive and there are also some very interesting national and provincial parks near Lethbridge Alberta. (waterton lakes, taber, dinosaur provincial parks) If you are driving in the winter it can be extremely slow going, however.
The east coast is also beautiful. It reminded me of the south of england, actually. If you have three weeks, I'd recommend driving from Vancouver to Toronto; its about 45 hours straight driving so you'd have to put in some solid driving days, but there'd be time to stop, camp, ski, sightsee etc. Also, you should remember that gas prices are increasing in Canada and driving is not as cheap as it was a few years ago ($1.10Canadian per litre). If you have two weeks, I,d recommend staying on the west coast and, besides heading to banff and jasper, going to whistler. If you like to hike there are some great parks in BC too.