As noncount nouns, they can be synonymous (e.g. room for the furniture= space for the furniture), but 'space' can also refer to 'outer space/the universe', and 'room' can, of course, also be a countable noun (as in 'my room').
Even when synonymous, though, there can be subtle differences: 'space' tends to be used literally (except, perhaps, in the popular expression 'personal space'), while 'room' is often metaphorical. Thus we might say
There's always room for new ideas.
!There's always space for new ideas.