FWIW, my wife and I dive with similar setups, BP + 36# Halcyon/Pioneer wings.
On our recent dives in Seattle, she dove with a FredT heavy STA, and I dove with the built-in adapter in the wing.
STA pros vs. built-in adapter:
1) More weight you can take off your weightbelt. (This is the primary reason for using the STA in our situation, since me wife needs around 22# of weight)
2) *Solid* tank/BP connection (when it works, see below)
3) Less 'fuss' when putting on the tank. It's a solid connection, and the tank/wing doesn't roll around at all. Sometimes the tanks shifts on the wing when diving if I don't cinch the straps down super tight.
STA cons vs. adapter
1) More weight. We're going on a warm-water trip in June, and the STA is staying at home, since we don't need it. Nor do we use it in our training dives in the pool
2) Something else to break/fail. During our trip to Seattle, the nut for the STA fell out on two different occasions, causing the tank/STA to separate from the wing/BP, which was 'interesting' to say the least. The first time it was the upper bolt, and this caused the regulator to get pulled out of my wife's mouth. The second time was less exciting as it was the bottom bolt, but trying to get on the boat wasn't much fun. The second time it happened when diving with Uncle Pug, so feel free to ask his opinion of the situation.
3) More stuff to carry to/from the boat. Having the wing with the STA on it is just more bulky. It's alot easier to stack/pack the dive gear without it on. (This is more important when travelling on airlines and such.)
With regard to the STA coming apart, I ended up stopping at West Marine in Seattle and buying longer bolts and lock-washers to (hopefully) prevent the situation happening again.
With the above background, I'm considering buying another STA for myself for use in cold water. I could use at least 2-3# of weight when diving in cold water, and if I had a heavy STA, that would let me use a bit more insulation, plus give me a bit more flexibility in how much weight I use in my ACB pockets. I dove with 23# in Seattle, and on the very last dive I added 1.5# ankle weights which allowed me to put a bit more air in my drysuit which seemed to help out.
I'm still not completely sure what I'm going to do. I may go the weightbelt route (like Uncle Pug does), but then again, if I can get the STA to stay together, I think I would prefer it to using a weightbelt.
So, if I were to summarize why I *like* the STA, it's because it provides weight without the need for a weighbelt. However, I have to admit that whenever I don't need the weight, I don't use the STA, despite the better tank/wing connection. I can live with a little bit of hassle.
Hope this helps!
Nate