How is each type of video camera stabilizer different?
There are a very wide and diverse range of different techniques for stabilization.

At the most basic, you have pure software stabilization.  This uses no specialized hardware, but rather tracks objects within a scene and then clips the video such that it maintains a frame which it can keep more stable.  This has an apparent impact of subjects moving less, but often produces an awkward feeling wavy effect as motion blur is still present in the source footage and occurs while the subject doesn't visibly appear to be moving.  It can also be used in support of other physical techniques.  It can handle minor corrections, but isn't advisable for large scale correction if it can be avoided.  This can be applied either in post or directly during capture on the camera (electronic image stabilization.)

At the next level, we move up to optical image stabilization systems.  There are two main techniques here.  The first utilizes a motion correction element in the lens along with small gyroscopes in the lens to keep the orientation of the corrective element relatively static.  This results in a correction for shake in the lens' orientation, but doesn't correct for lateral movement.  A similar system can be accomplished in camera by moving the sensor itself to correct for movement of the camera.  Gyroscopes feed the data on how to move the sensor.

Moving past that, the next big step up is a hand held gimbal stabilizer.  Gimbal stablizers work using inertia to offset movement rather than gyroscopic forces.  A typical handheld gimbal stabilizer consists of a rod with a free floating gimbal about 1/3 of the way along the length.  The camera is mounted on the top and a series of balancing weights are added to the bottom.  The inertia of the combined and properly balanced system results in orientation changes being absorbed at the gimbal as the inertia keeps the camera rig pointed in the same direction.  These provide far better stabilization for orientation and also provide a fair bit of stabilization for lateral movements, however they are considerably more expensive and more difficult to use well than an optical image stabilization system is.  It can also still be used with the previous mechanisms.

The next step up from this is a steadicam like rig.  Steadicam utilizes a gimbal stabilizer, however, rather than hand hold the gimbal, it is mounted on a spring arm attached to the operator's chest.  The spring arm allows for much better weight support as well as better absorption of lateral movement, thus it is able to deal with situations such as an operator running while operating the camera.

The final upgrade is to move back in to active systems again.  The inertial systems are all passive and thus don't require independent power (the OIS systems are active, but the inertial ones are passive.)  By adding high powered gyroscopes to an inertial system, additional stabilization can be achieved by providing active counter-action to changes in orientation.  The upside is that these hybrids provide the absolute best possible stabilization, but the downside is they are generally heavy, expensive (thousands to tens of thousands of dollars), power hungry and complex.