= Solution
A <symplectomorphism> is a <diffeomorphism>, so it preserves the <genus of a surface>. It also preserves the <orientation> determined by the <symplectic form> and, by integration of a <pullback of a differential form>, preserves the <symplectic area>.
Conversely, equal <genus of a surface> gives a <diffeomorphism> $f:\Sigma_0\to\Sigma_1$. We can choose $f$ to preserve the <orientations> determined by $\omega_0,\omega_1$: if necessary, compose with an <orientation-reversing diffeomorphism> of the standard <closed orientable surface> of that <genus of a surface>, obtained by reflection of a symmetric handle model. On $\Sigma_0$ put $\eta=f^*\omega_1$. Both $\eta$ and $\omega_0$ are positive <volume forms> for the same <orientation>, and their integrals agree. By <top-degree de Rham cohomology>, $[\eta]=[\omega_0]$.
The family $\omega_t=(1-t)\omega_0+t\eta$ consists of positive <volume forms>, hence <symplectic forms> in dimension two, with constant <de Rham cohomology> class. <Moser theorem> supplies $\psi$ with $\psi^*\eta=\omega_0$. Consequently
$$
\boxed{(f\circ\psi)^*\omega_1=\omega_0.}
$$
Therefore \b[the <genus of a surface> and <symplectic area> classify <compact> <connected> <symplectic surfaces>].
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