Agree with Sapphire Valley and Lake Lure recommendations. We live within 20 minutes of the Biltmore Estate and have several friends who work there. Very familiar with the area.

Besides the wonderful
Biltmore Estate or the historic
Grove Park Inn (open to the public), there are many fun activities in the Asheville area. Try tubing or canoeing down the French Broad River. Drive along, picnic or hike from the Blue Ridge Parkway (south are gorgeous views within 15 min, north includes the Folk Art Center with excellent mountain crafts and art pieces). Take in an Asheville Tourists baseball game at McCormick Field - they're a farm team for the Atlanta Braves, tickets are very reasonable. Area vendors (ex.,
Fun Depot) offer go-carts, night-glow laser tag and bowling, paintball as well as more typical amusements. Outside of Asheville, try some whitewater rafting - best places are west. About an hour west of Asheville, visit the
museum in Cherokee, where they also have an excellent living history Oconaluftee Village, a casino resort and "Unto These Hills," an historical outdoor drama in the summertime about the Trail of Tears.
Vacation Club Villas in Asheville is only ~10 min drive to the entrance of the Biltmore Estate. So it's fine for a base, for sleeping and maybe playing a little, hilly golf at the resort but there's not much there onsite for families. II only.
Sapphire Valley in Cashiers, NC has good golf, nice indoor and outdoor pool areas, fitness center, table tennis, gem stream, fishing, mini-golf, track&field, winter ski lift and warm season boat rentals at a smallish lake with a very pretty view. Many, many waterfall hikes in the area and it's about an hour's drive north to Cherokee attractions or casino or to Ghost Town amusements or whitewater rafting. Several resorts within 2-3 miles, as mentioned above, share amenities here and most require a car to get to the recreation centers.
Sapphire Valley resorts are ~1.5 hours drive on winding roads to the Biltmore Estate and downtown Asheville. II or RCI.
Lake Lure, NC has several very nice golf courses, good indoor and outdoor pool areas, fitness center with spa services, mini-golf, tennis courts, fishing, warm season boat tours and boat rentals on a decent-sized lake with pretty views and movie-making history. Again, several resorts within a couple miles share amenities here and a car is needed to reach the onsite recreation center from all of them. There's more golf here. It's about a 1.75 hour drive to Cherokee or Ghost Town from here but only ~15 min. to
Chimney Rock State Park with waterfall and other hikes.
Lake Lure resorts are about an hour's drive, maybe a little bit more, on winding roads to the Biltmore Estate and downtown Asheville. If you Mapquest it, note that when you come from Asheville to Lake Lure, you'll pass through Lake Lure's little town (and by a large grocery) and travel around the lake along winding roads for another 15 min. before reaching the resorts. II or RCI.
The Gatlinburg/Pigeon Forge, TN area is about 2.5 hours drive away from Asheville, not considering high traffic periods. Plan longer if your drive between Gat/PF and I-40 is particularly long or will be during heavy traffic periods, like Fri/Sat/Sun during summer, leaf season or UT football home games. If you want to stay either one of these areas and visit the other during your vacation, I strongly recommend getting a room for a night near the 2nd destination so you will enjoy your visit more. If you visit Gat/PF, don't miss the Smoky Mountains National Park! Enter in Gatlinburg proper.
Traveling between Gat/PF and Asheville, you'd pass exit 27 on I-40 that takes you into Maggie Valley, NC, a gorgeous mountainous area with a timeshare on a golf course with a pool (drive to rec bldg). Seems pretty well-maintained for an older resort with smallish rooms, similar to Vacation Club Villas in Asheville. Neither of these two resorts are as nice as Sapphire Valley or Lake Lure in resort amenities or rooms. But if you want to visit both Asheville and Gat/PF in the same week, it may be an option due to location.
Maggie Valley is only 20-25 min. drive to Cherokee, Ghost Town and whitewater rafting locations, as well as being ~45 min. to Asheville, ~1.6 hrs to Gat/PF. II only, I think.
For the nicest rooms and resort amenities, especially with teenagers, I'd go with Lake Lure or Sapphire Valley. If you're traveling during peak summer when schools are out, I'd choose Lake Lure for the location and great summertime resort activities. If you're traveling in shoulder season or you plan to try activities in Cherokee or rafting, as well as Asheville, I'd choose Sapphire for the amenities and location. HTH!