The "wet pipe" has water injected into it, usually at the header. This keeps exhaust gases cooler and tricks the pipe into thinking it is longer. Advantages: cooler running, normally less noisy and overall shorter length so in many cases the entire pipe can fit inside the hull. The pipe itself stays "cooler" to the touch. Disadvantages: typically makes less power than a dry pipe in part because of cooler exhaust gases.
The Wet Pipe is more of a sport boat pipe.
The "Dry pipe" does NOT have water injected INTO it...it may or may not have water cooled flanges, but this water is not intended to go inside the pipe itself: Advantages: typically much better performance than the wet pipe. Disadvantages: typically longer than the wet pipe, if both pipe are identical, the dry pipe will be louder. The pipe gets VERY hot...additional clearance and ventilation in the hull is required.
The Dry pipe is the pipe of choice for racers and for all out performance.