Yellowstone is much bigger, with a bigger variety of things to see. Geysers, 'paint pots', the canyon, the falls, lots of wildlife, etc. Plus you can camp there more easily because it's bigger so it has more campsites.
Yosemite is 4 hrs from Los Angeles and 4 hrs from San Francisco, so it tends to be crowded. You usually have to make reservations MONTHS in advance to camp there, on the valley floor. (If you like backpacking, though, you walk for 30 min. or an hour up a trail and you're all by yourself.) The mountains are incredibly beautiful, but the things to see are not as varied as Yellowstone.
River rafting, I think California has more opportunities. Not in Yellowstone, but close, in the foothills of the Sierras, on the American River (where gold was discovered).
Mountain climbing, I'd have no idea. The Rockies and the Sierras are both made of granite, which is good for climbing.
