Here is a response I was given when I ask the same question. Thanks to Travis of Stillwater:
Here's a quick Crash Course in Corallimorphians
Discosoma: The genus used to be called Actinodiscus, so you'll hear that too. These are generally the regular smooth mushrooms you see everyday, but a few can be kind of bumpy.
Rhodactis: These have protruding vesicles, but they branch. Often called Frilly or Hairy mushrooms because of this.
Amplexidiscus: Very similar to Rhodactis, but a little smoother. They grow to the size of a dinner plate and can eat fish and invertebrates
Ricordea florida: Vesicles are smooth and rather uniform across the body. There are no vesicles around the mouth. They tend to have elongated vesicles just around the rim of the body.
Ricordea yuma: These have smooth, protruding vesicles that grow to different sizes and lengths across the body. A ring of vesicles surrounds the mouth.
While there are many species of Discosoma, Rhodactis, and Amplexidiscus, there are only two Ricordea; R. yuma and R. florida