If there is a significant snow in the area, how hard will it be to travel the roads out of Cades Cove? Do they clear the roads soon after it snows?

Yes

They will close the gates if there is enough snow

The road/gate up the mountain will close. They will clear the roads as soon as possible. However, the severity of the snow storm will dictate the ability to clear the roads quickly.

Yep more likely to close for clearing