The air gun contains a cylinder with a plunger in it. You "charge" the cylinder with a pump, much like a bicycle pump does. It then has a "static" pressure when the plunger is fully extended.
To "cock" the air-powered gun, you place the spear in the barrel, place a T-handle over the tip, and force the spear into the barrel. This depresses the plunger and further pressurizes the gas in the chamber. The plunger/spear combination "locks" with a sear assembly just like a band gun does. When fired, the compressed air charge behind the plunger propels the spear forward.
All things considered, the band gun has more flexability (since you can choose your power easily depending on how many and what bands you load), and they are generally larger and thus more powerful. You can also load them progressively - that is, if there are three bands, you load one at a time. With a pneumatic gun you must load the entire pressure at once, which limits its ultimate power, and you must be able to reach the end of the spear with the T-handle while the gun is against your body in order to load it, which limits their maximum length (and thus power) significantly.
However, the pneumatic gun has an advantage in that there is pressure on the spear all the way down the barrel; that is, the plunger is under STATIC pressure as well as dynamic. This is not true for the band gun. The downside is that since this is AIR pressure, as you go deeper the gun gets less powerful! A pneumatic gun is extremely sensitive to the condition of the spear as well; if it is even slightly out of true, it may jam in the barrel when the gun is fired. A band gun will not shoot straight if the spear is bent slightly, but it will at least fire cleanly.
Note that there is no way to "unload" a pneumatic gun except to fire it. This is not true of a band gun - you can unload the bands without shooting the spear. As such if you keep your gun loaded on the way up (potential predator deterrent) you will have to discharge it behind the boat (while submerged!) before boarding if it is a pneumatic....
In general most "serious" hunters seem to use band guns, at least in the US. Pneumatics are popular in Europe, but I don't know why, specifically.
