How does light from a beauty dish behave, and when and why would you use one?
A beauty dish gives a softer light than an unmodified flash, but not as soft as a softbox or shoot-through umbrella.
This means that you can control the light more than when using a softbox.
A beauty dish is often used in portrait work, and can be flattering when used as a light on the subject's face, where it can give a nice balance between even softbox type illumination and harsh ringflash.
To my understanding, a Beauty dish is not really about making the light softer. That's what softboxes / bounce are for.
The true value of a beauty dish is that it focuses the beam of light in a 3D point/zone, thus simulating a virtual light at that point. This virtual light has the same property as a real one.
Getting this virtual light right in front of the face of the model makes the shadows on the side of the face more visible.
Making the edge of the face darker makes the face look slimmer visually, thus the "makes you look 20 years younger" reputation of the beauty dish.
The tricky part is to place that virtual light right, with no other information than how it is lighting the subject. Also, the "focusing" ability of the beauty dish is not perfect, thus the virtual light is not just a point light but more something like a tiny softbox.
If the beauty dish is too far away from the subject, it loses its point completely as it is equivalent to a simple harsh light (bare flash). If it is too close, then it will just produce a weird light, something close to what a ring flash would do. In both cases, that's a big waste of money (actually light focusing beauty dishes are damn expensive).
In general, the beauty dish will cause an even spread of light around the subject that provides for better contrast. Basically, it becomes less directional, illuminating from several directions rather than from above or to the side. So, in that sense, it would be used for portrait work or even product or macro. It is, however, more harsh than using a softbox setup, so be aware of the impact that can have on skin flaws and such.