What's the difference?
With thanks to Filip Wnek for the idea.
Here, we are looking at the meaning of room/space in the sense of an area that does not have something in it:
I need a space to park my car.
Is there space to park my car?
I haven't got room for all your junk!
Now let's start with the definition of room. This is an area that you can put things in. But this area may already have things in it like a cupboard, some boxes, a wardrobe, and so on. Here's an example:
Is there any room in your wardrobe to put my things in? (Is there any available area in your wardrobe to put my things in as I assume/see that it already has things in there and I'm not sure if I can get all my things in your wardrobe.)
Do you have any space in your flat? (Do you have, in your flat, an available area that I can use and that is not being used or occupied by someone/something else?)
But sometimes room and space can be swapped around but without making any real change to the meaning:
There is no room on the shelf for my books. (There are a lot of things on the shelf, leaving not enough available area to put all of my books, although it is possible some of my books can go there.)
There is no space on the shelf for my books. (There isn't any available area to put any of my books on the shelf because it is full.)
Indeed, only grammar and vocabulary pedants would really complain if you used the words wrongly; we natives will understand you. But for written work and tests, it's good to remember these very subtle differences.