One can define several maps between given structures in order to portray similarities or differences. The concept of isomorphism displays a deep structural overlapping, while for instance an homomorphism carries a weaker, one-directional mirroring.
Let us now take two first-order structures for a signature $S$, say $\mathfrak A$ and $\mathfrak B$, such that $\mathfrak A \cong_p \mathfrak B$ (i.e. structures are partially isomorphic): if the first-order signature $S$ is relational, $(\mathbb Q,<)\cong_p(\mathbb R,<)$ is an example of two partially isomorphic structures. Given the case that the domains of $\mathfrak A$ and $\mathfrak B$ are countable, if $\mathfrak A \cong_p \mathfrak B$ then $\mathfrak A \cong \mathfrak B$ (there is a famous theorem to prove it).
When we introduce into the picture the concept of two structures being finitely ($\mathfrak A \cong_f \mathfrak B$)or $\omega$-isomorphic (that is, $\forall n\leq \omega$ there is a chain of sets of partial isomorphisms $(I_n)$ such that $\mathfrak A \cong_\omega \mathfrak B$) we have that: $\mathfrak A \cong_p \mathfrak B$ $\Rightarrow$ $\mathfrak A \cong_f \mathfrak B$ but the converse isn't (usually) true.
Could you please provide an example of two elementary structures that are finitely isomorphic but not partially isomorphic? Thank you for the support.