When used in this sense, as an UNCOUNTABLE noun, yes, they do have the same meaning. 'There isn't much room in the office' and 'There isn't much space in the office' mean the same. If you are speaking literally, 'I need more room' and 'I need more space' have the same meaning - in the sense of physical area. But note that when we are speaking in non-literal terms, only 'space' can be used. For example, you might say to your boyfriend 'I need more space' in a psychological sense - 'room' wouldn't work here. However, this is the only usage of these words where they are interchangeable. When used as countable nouns, they have have different meanings. For example, you can say 'Leave a space between each chair', but you can't say 'a room' here. 'A room' - obviously - has a very different meaning.

I think "space" is the most common