what exactly are the characteristics of a beauty dish?
Harder light than from a regular softbox. Possibly more focused light than you would get from the little softboxes made for small flash, definitely more focused than from a studio softbox. Small enough to put almost in the model’s face, but keep it out of the photo. Large enough to deliver some wrap to the light. The parabolic reflector is supposed to focus the light on some distant point, well behind the model.
I don’t know, but it’s not a stretch to imagine some photographer saw a movie set and liked what a spotlight did to a model. Later the beauty dish was worked out as a way to get light from a strobe that had the characteristics of the movie spot light. The other place we sometimes see the same construction is some car’s high beam headlamps.
You might like this video: http://www.youtube.com/watch?v=w-cQyJWNvLY