Advantages of shooting off the shelf: Simplicity. Durability. Gets the arrow closer to the hand so it makes it easier to point and shoot rather than taking deliberate aim (although there is no law that says you can't take deliberate aim while shooting of the shelf ).

Advantages of using an elevated rest. Easier tuning. Sometimes allows more leeway in arrow spine selection. Better fletching clearance. Sometimes = increased arrow speed. Capable of better accuracy and consistency (more conducive to deliberate aiming techniques than for instinctive/poke-n-hope shooting). Also allows the use of plastic vanes, if you're so inclined.

For tournament shooting, where I needed the best accuracy possible, I nearly always used a bow with an elevated rest. For hunting, where the last thing I needed was a broken rest, I almost always shot off the shelf. Also, there are some 3D classes/tournaments that outlaw elevated rests.

I ALWAYS shot off the shelf for bowfishing. No rest hanging out there for the line to get tangled on.