The assumption implies that the natural embedding induces an isomorphism $\Gamma(X,\mathscr O_X)\simeq \Gamma(\overline X,\mathscr O_{\overline X})$. Then this means that  the complement of $X$ has at least codimension $2$ and *if* $X$ is normal, or at least $S_2$, then the same condition holds for every *reflexive* sheaf.

So, *if* $\Omega_{\overline X}$ is a reflexive sheaf, then the restriction 
$\Gamma(\overline X,\Omega_{\overline X})\to \Gamma(X,\Omega_{\overline X})=\Gamma(X,\Omega_{X})$ is an isomorphism.

More generally, let $Z:=\overline X\setminus X$. If $\mathrm{depth}_Z\Omega_{\overline X}\geq 2$ then the restriction $\Gamma(\overline X,\Omega_{\overline X})\to \Gamma(X,\Omega_{X})$ is an isomorphism.
This certainly holds if $\Omega_{\overline X}$ is a reflexive sheaf and $X$ is $S_2$, but obviously could hold "by accident" even if one of those conditions fail.


**EDIT 1** removed intro paragraph about the starting assumption.

**EDIT 2** added "more generally" paragraph