Both '실' and '방' are from Chinese: 室 and 房, respectively. There may be some nuance to their usage but I believe they essentially mean the same thing. However, '실' is a bound noun and cannot be used by itself. Rather, it appears as part (generally on the end) of other words to indicate a room where something is done. On the other hand, '방' can be used by itself to mean 'room' and is far less frequently attached to the ends of other words.