A biased summary
Advantages: none over a top quality clincher (normal tyres) *UNLESS* you are running carbon rims.
Disadvantages: you have to carry an entire spare tub around with you (or two if you expect two punctures) so you can put a new one on when you puncture. Tubs can't be (easily) fixed when punctured so it can get expensive.
Back in the day, tubs were much better than clinchers for racing: lighter, lower rolling resistance, cornered better etc. That isn't true anymore and they were dying out.
The thing that has bought them back is that you can make a very light deep section rim in carbon (and even make it cheap) and run tubs with it. Until very recently running clinchers on a carbon rim meant adding an alloy tyre bed to the rim, thus negating the light weight. You can now get a vrey few carbon clincher rims but they are hard to make and thus expensive.
Summary: I'd be tempted by a pair of deep sections + tubs for racing where I don't have to worry about a spare. Something like these:
http://www.planet-x-bikes.com/triathlon ... on=282:282
Wheels like this can be lighter than standard wheels but more aero as well. Win, win.
Since I nearly always ride out to races however, I can't really justify them. You also need different brake blocks for carbon rims if you want any braking power in the wet.