Why use a rebreather? Several reasons: One, there aren't any bubbles. Photographers like this, because they feel they don't scare off marine life as much. Some people, like my husband, just hate the sound and feel of their exhaust bubbles.
Second: By rebreathing your exhaled gas, you can get far more time off a given gas supply. Many people diving closed circuit rebreathers (the kind that don't exhaust any of the gas unless told to do so) can use very small (like 19 cf) tanks. When you are diving expensive gases, like helium, this becomes quite advantageous.
Third: Minimizing decompression. Because closed circuit rebreathers can continuously monitor and vary the oxygen percentage in what you're breathing, they can keep the inhaled nitrogen as low as is safe for any given depth. This results in shorter decompression times for technical divers.
Recreational divers DO use rebreathers, but they are expensive, require specific training for any given unit, and are extremely unforgiving of any carelessness on the part of the user.