Consider the short exact sequence $$ 0\to f^*\Omega_C\to \Omega_X\to \Omega_{X/C}\to 0, $$ and the long exact sequence induced by $f_*$: $$ 0\to \Omega_C\to f_*\Omega_X\to f_*\Omega_{X/C}\to \Omega_C\otimes R^1 f_* \mathscr O_X\to \ldots $$ If $f$ is isotrivial, then $\beta: f_*\Omega_{X/C}\to \Omega_C\otimes R^1 f_* \mathscr O_X$ is generically injective (at least where $f$ is smooth $f_*\Omega_{X/C}$ is a line bundle and $\beta$ is non-zero). If $\Omega_{X/C}$ is torsion-free, then so is $f_*\Omega_{X/C}$ and hence $\beta$ is injective everywhere. That implies, that then $\alpha:\Omega_C\to f_*\Omega_{X}$ is an isomorphism.
Therefore, $H^0(U,\omega_C)=H^0(f^{-1}U,\Omega_X)$ for every $U\subseteq C$ open.