Instead of an example, I give an existence proof:

Take any finite or countable language, for example the language of equality. Since all formulas (even in $\omega$-logic) are finite, there are only countably many formulas, hence at most $\mathfrak c:= 2^{\aleph_0}$ many theories. Find more than continuum many cardinalities (for example $\{\aleph_\alpha:\alpha < \mathfrak c^+\}$), and for each such cardinality $\kappa$ find a structure whose size is $\kappa$. These structures are pairwise non-isomorphic, but there must be two that satisfy the same $\omega$-sentence.

(For languages with $\lambda$ many symbols, replace $\mathfrak c^+$ by $(2^\lambda)^+$.)

Instead of an example, I give an existence proof:

Take any finite or countable language, for example the language of equality. Since all formulas (even in $\omega$-logic) are finite, there are only countably many formulas, hence at most $\mathfrak c:= 2^{\aleph_0}$ many theories. Find more than continuum many cardinalities (for example $\{\aleph_\alpha:\alpha < \mathfrak c^+\}$), and for each such cardinality $\kappa$ find a structure whose size is $\kappa$. These structures are pairwise non-isomorphic, but there must be two that satisfy the same set of $\omega$-sentences.

(For languages with $\lambda$ many symbols, replace $\mathfrak c^+$ by $(2^\lambda)^+$.)