The AT has more views, on that particular route you get Clingmans Dome, Newfound Gap, Charlies Bunion, and Mt. Cammerer Fire Tower (0.6 miles off the AT but a must see). The BMT has Clingmans Dome and Mt. Sterling Fire Tower.
The AT will be crowded (particularly between Clingmans Dome and Charlie’s Bunion) and you have to stay in shelters. The BMT will not be very crowded and you stay in designated campsites, so it somewhat depends on if you want the higher trail, or the lower trail, and if you want to see people or not. Some people like the shelters and others don’t, but if your on the AT, you have to stay in them.
For the AT, as for shelters, I haven’t stayed at Mt. Collins since it was remodeled, but it was one of the last GSMNP AT shelters to get the update, so it should be nice. It’s not a bad location though.
I’ve only stayed at Icewater Spring once, as it’s often a zoo being so close to the road. It’s not a bad spot either, and the spring is pretty much right in the trail.
Pecks is an older shelter, and a little ways off the trail, but it’s not bad. The water is down the hill a ways. Pecks is my wife’s favorite shelter.
I’ve stayed at Tricorner Knob more than any, due to it’s location. The water is right by the shelter, and it’s newish. I haven’t been there in a few years but there used to be a problem bear in that area. It never caused any real problems but would visit every evening.
Cosby is a newer shelter too and water is right by the shelter.
I’ve never stayed in Davenport Gap so I can’t comment.
My favorite part of the trail through there is between Charlie’s Bunion and Pecks Corner. It’s a really cool knife edge(ish) ridge in a few spots.
For the BMT what are you planning on taking from Clingmans to the BMT? Forney Creek? Forney Ridge? Noland Divide? I’ve stayed at CS#52 before. It’s been a decade but I remember it as being nice. #47 is small and right on the creek with a bear pole (the others all have cables). There is a bridge over Raven Fork now and one a Beech Gap. CS#44 is a horse camp, and was trashed last time I was there. Laurel Gap Shelter has been redone since I was there, but it’s in a nice open area. The BMT from Laurel Gap Shelter to Big Creek is very nice. It’s a nice ridgewalk most of the way with some decent views and a Fire Tower at Mt Sterling. CS#38 is a little small and rocky, and can be crowded, but for some reason I really like it.
Either way I would park at Big Creek – If your on the AT that means turning off just before Davenport Gap and taking the Chestnut Branch Trail (2.1miles).