I've read lots of stuff that it purports to be good for (and i work in a physio department so people there are pretty much brainwashed on one view of how to train) - but what about the bad side/limitations - is there one?
Reduces total load able to lift. Reduces intensity.
Basically you'll get stronger faster bench pressing compared to DB pressing on a swiss ball.
Stability ball training is great... once in a blue moon. The funny thing to me is while it is touted as "functional" strength, I think the biggest problem with it is that it is very unspecific for most people. Unless you plan on bench pressing aboard a rowboat in a storm, I would only break out the Swiss Ball once every other month or so.
Stability training requires more muscles per lift as opposed to, "regular strength training." Perhaps it burns more calories during training, and can improve muscle imbalances and weak links.