There was a post in the Tech Talk Forum just about 2 weeks ago about Beauty Dishes. We posted sample images and everyone gave their 2 cents about which they liked better (compared to other light forms, softboxes, etc) See if you can look back and find that post.
I have a 24 inch Photogenic Beauty Dish. It's like a big aluminum pasta bowl with a little 3 inch insert in the middle. The light from the bulb hits the insert and then bounces all around the dish, and then spills out onto the model.
I can remove the insert so some of the light goes straight at the model while a bunch of it bounces around the dish and spills out at the model. It's obviously much "hotter" with the insert removed. Some places make, or you can make your own "diffuser sock" that goes over the whole thing made of sheer fabric so the light bounces around the dish and then as it spills out has to go through the fabric for even more diffusion.
Why use it compared to a light box or umbrella, or flat panel, or small pan or snoot.....well, for the same reason they make snoots, small pans, flat panels and light boxes.
They each have their own characteristics.
My beauty dish provides a nice poppy yet diffused light source that I really like. It's "harder" than a softbox, but not as hard as a bare bulb (ovbiously). Compared to a shoot through or bounce back umbrella, it is more distinct.
It's not as harsh as a ring light but close.
The dish I have for my Photogenic Powerlights was under $200.00 and I'm quite happy I bought it. At that price it was actually cheaper than some of the bigger softboxes I have.
On the other hand, I have seen them that attach to my Profotos or other brands that are six or seven hundred dollars....I'm not sure I want one that bad.
I've seen photos of dishes that are substantially bigger like 30 or 36 inches. I bet one of those would be cool.