We have a cover over the one we use but if it gets really rough outside we will store it in the building outside. The one we have is a about 12 X 15 or so and we keep other stuff out there as well (it is the working shop so the grill feels at home with the other tools and gadgets).
We get a lot of snow in my neck of the woods too. We really should store the grill in the shed or garage, but there just isn't room, so it gets left outside. Surprisingly, it doesn't really effect it too much.
Winters are not that bad here so mine stay outside. It is the hurricanes you have to worry about the smoker gets put away as that is the only one outside. But then again we haven't had a hurricane in forever.
Our grill has a cover, but since we often have a snowy winter around here, we keep it inside our mini-storage area in our backyard. Our grill is quite big and occupies space, but we manage to fit it in the storage area.