The park service in recent years has become very proactive regarding road conditions on U.S. 441/Newfound Gap Road. In my recent experience, having traveled to the area in early January the past three years and following road reports on their Twitter account, if there's any remotely significant accumulation, they close the road until it can be cleared.
Basically, if it's dangerous, it will be closed and you won't be able to drive it.
Reid is correct. If there is snow they will close off the gates to Newfound Gap Road. I would watch the local weather reports when you get in town and check for park updates on Twitter. It wouldn't hurt to stop for info at Sugarlands Visitor Center.
As everyone else has stated, the park service Is always on top of the situation when it comes to snow removal or clearing up downed trees and/or fallen rocks on the road from storms. They will close the roads if any of these problems arise.
And remember, all the NPS can do is plow and sand; salting the roads is not an option in the national park.
I know 441 is the best way to go when you're interested in seeing sights along the way, but NC is still easily accessible during the winter via I-40. It takes me 1.25 hours to get to Maggie Valley/Waynesville from Pigeon Forge. From there, Asheville is about 30 minutes, and Cherokee & Dillsboro are about 20 minutes.
That's not to say that 441 WILL be closed, even if it snows. We've had some good times in and around Newfound Gap in the snow.
Enjoy your trip!