I just did a trip last weekend where I hiked both the Rainbow Falls and Bullhead Trail. I hiked the Rainbow Falls trail to LeConte, then took the Boulevard Trail over to the AT, then went downhill to Kephart Shelter via the Sweat Heifer Creek Trail. The next day I took the Grassy Branch Trail to the Dry Sluice Gap Trail which led me back to the AT and the Boulevard Trail back to LeConte. I then returned to my car via the Bullhead this time.
The Rainbow Falls trail climbs a little over 4000 feet in 6.7 to get to the top of LeConte, it’s not that bad though, it is just long and steady. I finished the climb in 2:26 and that was after getting behind a few slower dayhikers and stopping at the Falls for a snack. There are a couple of nice falls and cascades other than Rainbow Falls itself, but the downsides are the number of people that are on the trail and the fact you have to drive through part of Gatlinburg to get to the trailhead
I was really impressed with the Bullhead Trail, it had quite a few quasi views where you can barely see over the roto and mountain laurel. It shares a trailhead with Rainbow Falls so it sees a lot of traffic as well, though not nearly as much as Rainbow Falls itself. Jonathan is right and if you choose to stay at LeConte (call and get reservations ASAP (but no more than 30 days before the trip) as it fills up quick ) you can get a great sunset and sunrise view just a few tenths of a mile in either direction.
Here are my pics from my trip if you want to see what to expect: