There's bumper plates, steel plates, calibrated plates, and so on. What other types are there, and what are each used/good for? Pros vs cons?
Bumper plates are for olympic lifting.
In general, any weight coated in rubber is 1) to dissipate the reaction when the plate hits the ground 2) aesthetics.
Aesthetics are much more prevalent with dumbbells. Gyms like to coat the DBs in rubber, as well as their logo. The tradeoff though, that I don't think many gym owners fully appreciate, is that rubber wears off, eventually exposing the metal. At which point, you pretty much need to buy an entire new rack of weights, which is very costly.
Lastly, you won't see powerlifters use bumper or rubber plates much. One reason for this is the rubber tends to take up more room. It's not a concern for many, but for some strong dudes out there, they start worrying about running out of room on the barbell (or leg press machine).