How do you explain the difference between 'place' and 'space'?

I’m an English teacher myself (to foreign students), but a student of mine asked me this and I wasn’t able to give a concise answer. Googling this came up with geography and philosophy results, when I just mean this in terms of grammar. I couldn’t find anything on this stack exchange either.

I think space is usually uncountable, but a parking space is an exception. A place could refer to a building or a seat on a train or a place on a course.

Mistakes between them could be: ‘I’m looking for a parking place’ and ‘There’s no place in this room’.

But how would you clearly explain the difference between the two?