Can't speak for Yellowstone, but here in New Hampshire we get a fair ammount of snow. If the snow covers the trail and gets packed down it's slippery as heck and you'll want studs. If it's unpacked you should be able to ride it with knobbies but it takes a lot of work to get through. If it's too deep then you're SOL, it's torture to ride in that stuff, kinda like riding in really, really soft sand (only the sand is slippery). If the snow is light and the trail is clear then it'll also be wet and muddy (that water has to go somewhere). Depending on the soil this can lead to trail erosion and places around here frown on mountain biking during what we call "mud season" which is spring thaw.

I do bike in winter conditions, but I greatly prefer summer conditions. I certainly wouldn't travel TO somewhere specifically to ride in snow. I'd much rather escape it.