Difference between "space" and "mathematical structure"?

Neither of these words have a single mathematical definition. The English words can be used in essentially all the same situations, but you often think of a "space" as more geometric and a "structure" as more algebraic. The best approximation to a general "space" for many purposes is a topological space, but Grothendieck generalized further than that, to what are called topoi.

In model theory a "structure" is a set in which we can interpret some logical language, which is to say a set with some distinguished elements and some functions and relations on it. Some of the most common languages structures interpret are those of groups, rings, and fields, which have no relations, functions are addition and/or multiplication, and distinguished identity elements for those operation. We also have the language of partially ordered sets, which has the relation $\leq$ and neither functions nor constants.

So you could think of "structures" as places we do algebra, and "spaces" as places we do geometry. Then a lot of great mathematics has come from passing from structures to spaces and vice versa, as when we look at the fundamental group of a topological space or the spectrum of a ring. But in the end, the distinction is neither hard nor fast and only goes so far: many things are obviously both structures and spaces, some things are not obviously either, and some people might well disagree with everything I've said here.