Toobliness
Main advantage is the fact you can run very low pressures with no pinch flats and thorns generally don't result in flats either! But despite what folks say, specific tubeless tyres are a lot heavier as they have much thicker sidewalls to prevent rolling/popping off the rim with the lower pressures. The alternative is to run something like Stans no tubes in standard tyres, but you cant run 20-28 psi as they'll pop or roll off the rim.
The internal rim tapes are quite a bit heavier too. I usually run a 450gram tyre, the equivalent tubeless, same profile section would be at least 650 with some up to 800. I personally run Latex inner tubes that weigh only 100gram and are very thorn resistant. Tubeless have a lot of advantages, but on the pro circuit are still less than 40 percent usage, despite being around for a while. High weight is the reson why, also the liquid latex sloshing about(even tho its only a small amount) in the tube can have a strange effects on handling.