You can drive the both of them from Edmonton in about the same time, give or take 20 minutes. For Banff, you wouldn't actually go through Calgary, but would cut West about 20 minutes north of Calgary.
Jasper would get boring after a day, unless you were doing some hiking along the trails or driving to the Columbia Icefields for a tour. Its only a couple blocks of actual restaurants and shops and whatnot. Its lovely, absolutely lovely, but small and more of a service station for people doing outdoorsy stuff in the Parks. Banff is much bigger and much more impressive, has more to do within the confines of that (still small) town, like the hot springs, Banff Springs Hotel, a couple museums, greater selection of restaurants and shops, a scenic drive up Norquay or to Lake Louise (45 minutes or so through scenic mountain roads), the waterfalls, and some easy hikes in the area. I won't diss Jasper - its fantastic. But its too small to spend 2 full days if all you're doing is wandering around the townsite and fucking. Go to Banff (turn West at Aridrie, work your way to Highway 22 then to Cochrane, then to the T-Can or highway 1), stay at the Springs or Rimrock if you can afford it.
