What is the difference between shock absorber and steering damper?

The main difference is that its valved differently. Street shocks are often times 60/40 with a soft damping for control and ride quality. Performance handling shocks are typically 60/40 to 70/30 with a tighter damping to control the stiffer spring rates often associated with handling suspensions. Drag shocks are usually 90/10 to promote weight transfer but are unsuitable for street use.

Steering dampers are always 50/50 and have damping that is matched to the steering system. Obviously a monster truck with 78" tires needs more damping than an 4x4 truck with 31" tires.

So that was the long answer. The short answer is: not much is different except damping rates, but you can't simply mount any shock to a steering stabilizer and expect it to work.