Here's why:
* Proper nouns are names of specific people, places, or things. They are capitalized. For example, "John", "Paris", "Amazon River".
* Common nouns are general names for people, places, or things. They are not capitalized. For example, "man", "city", "river".
"Coat room" is a common noun because it is a general term for a room used to store coats. It doesn't refer to a specific, unique room.
However, if a coat room had a specific name, such as "The Cloakroom" or "The Winter Garden", then that name would be a proper noun and capitalized.