Obviously, if it gets too deep snow can make it impossible to ride. But snow also provides a great opportunity to look at a familiar place in a new way, too. A familiar rocky section of trail covered in fresh snow can be a new challenge to navigate because you can't see the rocks under the powder, but that same trail a few days later can feel smooth and faster than usual once a lot of the snow has been packed down. Ice is the only real enemy you have to look out for - hitting the ground on an icy section is never fun.
Aside from snow, the only element of winter you have to adapt to is frozen extremities. But that's usually a temporary problem that goes away after a while. I usually find that after about 20 minutes on a really cold morning ride, I have to stop to get feeling back in my fingers, but once the blood starts flowing again, I usually don't have to deal with it again during that ride. Excruciating digit pain as they thaw out builds character, doesn't it?