I would say the biggest pro is that they don't create a strong current, which is good for bettas. And especially good for fry/juveniles/weak bettas and plants. You don't have to get a pure sponge filter to have a bubble filter. This kind is really nice, I'm enjoying it, and it has some really good filtration media:
You do need to buy a separate air pump and some tubing for it, and a flow adjuster if you want to be able to do that. But all things considered, I really prefer this now that I've tried it, and I will probably be switching over to mostly this type of filter. Way better than those Whisper filters that dump a current of water all the time, and they're noisy when the water starts to evaporate from the tank. If you shop around you could end up spending the same amount on an air pump with multiple outlets for bubble filters in different tanks as you would on separate filters for them all.