I dove the Shark dives with BAD in 2006 and with Aquatrek in 2012. I think that what factored most into the my choice at each time was where I was staying. As mentioned in fiji_diver's reply, Aquatrek is across the road from The Pearl Resort, which is where I chose to stay in 2012. In 2006, I stayed at the then-Laguna Resort, but had been shutdown in 2012. It is a bit of a trek from the main road, and that is where BAD was (is?) based. I also had troubles exchanging emails with BAD when I was making my arrangements for the 2012 trip. If not for that, I probably would have gone back to them, since they did pick-ups at The Pearl just the same.
Both operations were very good, and as fiji-diver noted, who was first is not really of relevance.
As also noted by fiji-diver, one difference between the two is/was that BAD went deeper on the first dive, but perhaps as a coincidence, those were the dives when we saw Tigers, including one big, naughty one called Scarface. In addition, when we dove with BAD, all divers were in one straight line on a shelf slightly above and behind the feeder. Don't know if that is still the case. Whereas with Aquatrek, the divers were allowed to form a U, but only those with large cameras (to use as protective shields) were allowed to be at either leg of the "U". If you were on one of legs of the "U", there was no wall to protect your back so your head needed to be like a turret, altho if you were looking for close in-your-face encounters, this was the place to be at.
You might consider diving with both if doing the shark dives is all you wish to do. Back then - don't know about today - each operation did the shark dives on certain days an not on others - maybe so that the sharks wouldn't get too used to just being hand-fed and forget about hunting on their own(?). So if that is the case, you could check and see which days one operation goes out and fill in the days that they don't with the other operation (assuming that they go out on those other days).