They really are very similar. If your family loves a road trip, then try Branson and see which you prefer. If the extra drive time (and gas) are a drag, I don't think the differences make it worth the longer drive, personally. Here are a few differences that I see, which may encourage a visit to Branson MO.
Branson has far more shows so more variety and selection. It seems to be easier to find discounted show tickets too. There are lakes nearby, if you like to rent a boat or go on a boat for a dinner show. Silver Dollar City has more thrill rides than it's TN sister, Dollywood (though Dollywood is catching up) and SDC seems to have a couple more live shows. Celebration in Branson has more typical amusement park rides for the evening. Downtown Branson's new riverfront district (by Bass Pro Shops) is pretty for an evening stroll and watching the fountains.
Going in favor of a return visit to Pigeon Forge/Gatlinburg TN, it has the Great Smoky Mountains National Park - beautiful year-round and pleasantly cooler in the hot summertime. We love the mountains and the MO Ozarks are not nearly as striking. The TN/NC area has whitewater rafting and lovely historic Cades Cove. If you visit Cades Cove, go early in the day and bring a picnic lunch, swimsuits and towels to wade in the river afterward - very popular. Cherokee is a half hour+ away for authentic Native living museums, crafts/shopping and casinos. Asheville NC's Biltmore Estate is a 2 hr drive and some people add an overnighter there to their TN vacation. We find it relatively easy to avoid the worst of the traffic in PF/Gat by using side roads and parallel routes. This is easier than avoiding the crawl-and-stop traffic on the Branson strip, IMO.
Both places have a lot of the same attractions, touristy museums, outlet malls, chain restaurants and nice resorts, cabins, rentals, etc. Our bias is heavily in favor of TN but we only live 2 hours away, compared with a much longer drive for Branson. So for us, Branson was a once-in-a-great-while trip, while we don't get tired of visiting PF/Gatlinburg and the Smokies, several times a year. HTH.

Have fun, whichever you choose!