My wife and I go down there quite regularly. Tennessee fishing requirements are a little daunting. There is a few camp grounds in and around the Cades Cove & Greenbrier area which is in the Great Smoky Mountain Nation Park (GSMNP).
If you want to fish the Gatlinburg you will need a Gatlinburg fish permit. With the Gatlinburg permit you can ONLY fish in the Gatlinburg city limits. If you want to fish inside the GSMNP you will need a GSMNP permit. The GSMNP permit permits you to fish anywhere inside the park, including North Carolina side of the park but ONLY inside the park.
There is two rivers/creeks and their tributaries you cant fish in TN & NC. In TN you can NOT fish and it is the Lynn Camp Prong and it's tributaries upstream from its junction with Thunderhead Prong. In NC you can NOT fish Bear Creek upstream from its junction with Forney Creek. These two creeks are closed due to the park letting the Brook trout repopulate these areas. Their numbers are down because of the acid rain that falls in and around the Smokies caused by our air pollution
The following link may answer some more fishing requirement question you have:
Great Smoky Mountains National Park - Fishing (U.S. National Park Service)
While you are in the area visit the two fly shops that I know of:
The Smoky Mountain Angler (Gatlinburg)
Gatlinburg?s Complete Flyshop and Fishing Guide Service
Little River Outfitters (Townsend)
Little River Outfitters - Fly Fishing in the Great Smoky Mountains - Online Catalog
The before mentioned shops sell fishing permit, among other retailers. The GSMNP, themselves, do not sell the fishing permits.
You'll love the The Smokies. Hope this helps.