I think that 部屋 usually refers to a room in a house/apartment, where someone resides. 室 is somewhat broader and often refers to a room with a particular function.
This seems to be reflected in the names of the rooms in a house/apartment:
Rooms with a particular function
- 寝室
- 客室
- 地下室
Rooms that are lived in
- 子供部屋
- 仕事部屋
- おもちゃ[部屋]{べや} or 遊び部屋
In other buildings, too, special-purpose rooms usually end in [室]{しつ}, e.g. 事務室, 図書室, 洗濯室, 音楽室. (Also cf. 和室 "Japanese-style room".) Maybe that is just because 室, which is 音読み, sounds more formal than 部屋, which is 訓読み. For example, a public building won't have any 部屋, only rooms with a description (like 演奏室) or a number.
If your rooms are numbered, they are usually numbered as 132[号室]{ごうしつ}, but you usually count rooms as [一]{ひと}[部屋]{へや}, etc.
Note that 室 is a bit broader than the English "room". 132号室 could be a small apartment, consisting of several rooms. I think of 室 as anything that has one door from a hallway.
One last remark would be that I talked about the difference between 部屋 and 室 as suffix. But in conversational Japanese, I would say that 室 works almost exclusively as suffix (or prefix), whereas 部屋 is a word that can be used by itself to refer to a room. So if you want to say "I entered the room", or anything similar (i.e. without specifying which room), your choice will be limited to 部屋 (unless you want to sound literary).
