Mooshrooms are similar to Cows, but they only spawn in mushroom biomes. Similar to normal cows, a Mooshroom can give milk if milked with an iron Bucket. Unlike cows, a mooshroom can give Mushroom Stew if a wooden bowl is used as opposed to a bucket. When killed, a Mooshroom will drop raw beef and leather like a normal cow, making them useful for farming as Mooshrooms have a differing world limit than cows..
Mooshrooms can be turned back into normal cows by shearing them which will give you 2-5 red mushrooms.
The mooshroom babies cannot be sheared and will not give you mushroom stew right away, but it can be milked right away unlike normal baby cows.
A Mooshroom will give mushroom stew even though it has no brown mushrooms, though both brown and red mushrooms are essential for crafting mushroom stew.
Mooshrooms are likely the rarest passive mob in the game since not all worlds have a Mushroom Biome.