The typical protractor-style compass is an optical compass. To shoot an azimuth with it you align marks on the compass and read the bearing off the dial. The other varieties have a sighting wire which is placed over the target and a separate mechanism to help reading the bearing. The primary advantage is accuracy because the bearing can be read while observing the target; the primary drawback is that they require a separate protractor to take bearings off of a map.
A lensatic compass has a lens and wire which can be unfolded. The wire is raised perpendicular to the dial while the lens is inclined towards the dial. With the compass held to your eye you align a notch on the lens holder, wire, and target much like iron sights on a rifle. Then by looking through the lens you can read the bearing off the dial. Because you don't need to move the compass to see the object and dial it's much more accurate than optical types.
The prismatic compass is similarly constructed, with a prism rather than a lens. It functions much the same as a viewfinder in an SLR camera. You look through a hole at the target object and the prism reflects the dial into the bottom of your field of view. This eliminates the need for careful alignment of the lensatic style; you don't even need to move an eye to glance at the lens.
For hiking and camping the type is completely irrelevant because the margin of error in dead reckoning grossly exceeds the error of any sighting compass. For precision work from a fixed position such as a surveyor's tripod or fire tower a prismatic compass is the natural choice because it's a few minutes more accurate than the lensatic. The military tends towards lensatic models because they're accurate enough to direct fire, cheap, and durable.
Don't overlook the durability angle if you've already decided to forgo the integrated protractor/measurement of a baseplate compass. The military ones are housed in a steel clamshell which takes beatings much better than plastic models. There is something to be said for knowing you won't bust your compass should you happen to fall on it.