I grew up about 50 mi north of Asheville (68 mi by roads). There's not a ton of real "tourist" attractions - but the overall geography is beautiful. Biltmore is a very fun half-day and is quite impressive. It's a little north of the direct route from Nashville to Kiawah, but if you want some serious local flavor, check out
The Carter Family Fold, a Bluegrass treasure founded by June Carter Cash's uncle(?), I believe. The quality of the acts can vary greatly - from Johnny Cash on down. My high school biology teacher played banjo in a band that gigged there regularly, and their bass player has now won many Grammies playing with Alison Krauss.
Be forewarned if you go to the Smokies in the summer, though. It gets nearly as many visitors each year as Yellowstone, Yosemite, and the Grand Canyon - COMBINED. It's one of the few big National Parks near eastern population centers, and it shows. Traffic can crawl at 5-10 mph during daylight hours. The park itself IS more beautiful than the surrounding area, because it has the tallest mountains and is less developed, but it's really just a difference of degree. If you want to go on some really nice hikes instead of sitting in traffic, I'd look at Roan Mountain (the only point where the Appalachain Trail goes above 6,000 ft) and Mount Mitchell in NC instead. If you go to Dollywood, I'll never speak to you again.
If you enjoy driving (and I mean really driving), check out
Deal's Gap - 318 curves in 11 miles. Extra points for taking out motorcycles with sidecars... In that vein, there's also the Bristol International Raceway, the only NASCAR track completely surrounded by a stadium (1/2 mile).
There's some great whitewater on the Nantahala River (used for the Atlanta Olympics whitewater events, I believe) if that's your thing.
That's about what I can remember from 20 years ago, when I moved from the area - have a great trip!