Disadvantages: can contain unwanted pests (that's a big one), more expensive than dry rock, and depending on the rock available may not be as porous meaning it won't hold as much beneficial bacteria as other rock.
Advantages: your tank will cycle quicker, beneficial bacteria, may have coralline algae already growing on it (purple colored rock)
Perks of dry rock: cheaper around $2.59-3.99 a pound. Clean slate. No pests or possibilities of introducing algae, much easier to work with scaling whether you want to use aqurium safe concrete that takes a couple of days to cure (be safe to go into the aquarium)
There are different types of rocks. Like i said some more porous and you'll be paying more for a live rock that's porous due to all the water inside so your basically paying a percent for water. All my tanks I start dry rock pukani from bulk reef supply its $3.79 a pound and you can tell them what size pieces you want in the notes. But you don't get to choose and see the pieces in front of you. Best part about this rock it's light very porous so a 5 pound rock will be the size of a football where a 5 pound Carib Sea rock will be the size of your hand. It's very easy to aquascape because you can use a saw and just cut it and do basically what you want with it.