Structures and their substructures.

Question

Let $S_1, S_2$ be structures. Let $S_1 \equiv S_2$ Can we say something interesting about their substructures?

Answer 1

Not much. Certainly every finite substructure of $S_1$ is isomorphic to a finite substructure of $S_2$, and vice versa (because a finite structure can be completely characterized by a formula). Provided both structures are infinite and realize at least one type in common, they also have a common infinite substructure (up to isomorphism) constructed by taking the structure generated by a fixed element of the common type. Beyond that, there isn't really anything else to say, so far as I know.