I imagine you'll be taking U.S. 441 through the Park to Gatlinburg? It's a nice drive, albeit with a fair amount of traffic. Great views, though. Cades Cove is along the way too.
If you have time, I'd recommend a ride on the
Great Smoky Mountains Railway. Their Nantahala Gorge trip is nice. A decent ride, not too long (about 4.5 hours), with a lot of beautiful scenery. And the open air cars offer a better experiance, in my opinion.
There's
Unto These Hills, that the Cherokee Indians put on, but the last performance is August 20th, so you might not be able to make that. Definately worth it if you can, though.
Oconaluftee Indian Village is nearby too.
If you're into dams, go by Fontana Lake & Dam. Tallest concrete dam East of the Rockies.
As garrettwc said, the Biltmore Estate is a must see. They just opened the fourth floor of the house for the first time. There's also the
Thomas Wolfe House in Asheville.
The highest point East of the Mississippi,
Mount Mitchell, is about 40 miles Northeast of Asheville. Lots of dead trees, but a decent hike and great views on a clear day.
On past Asheville, on your way to Charlotte, is
Chimney Rock Park. Never actually been up on the rock myself, but pretty cool from what I hear.