$\DeclareMathOperator\Ring{Ring}\DeclareMathOperator\Aff{Aff}\DeclareMathOperator\op{op}\DeclareMathOperator\Sch{Sch}\DeclareMathOperator\Zar{Zar}$The category of schemes sits, fully faithfully, in various categories of sheaves on $\Ring^{\op} =\Aff$ (for the Zariski topology, etale topology, etc.) Are there epimorphisms in $\Sch$ which are not epimorphisms in the larger category of sheaves on $\Ring^{\op}$ for a given topology? Are there useful criterions which tell me that a epimorphism in $\Sch$ is also a sheaf-epimorphism of the associated functor of points?

More specifically, are there categorical quotients of a scheme $X$ by an internal equivalence relation $R$ on $X$ such that the quotient map $p:X\to X/R$ is a quotient in $\Sch$ but not in for example the big Zariski topos?

Note that the categorical quotient $p:X\to X/R$ in $\Zar$ always exist (because it is a topos). If $X/R$ happens to be a scheme, then $p$ is also a quotient in the category $\Sch$. But if $X/R$ is not a scheme, then there might still be a different map $X\to Y$ which satisfies the weaker universal property in $\Sch$ but not in $\Zar$. Does this happen and do such quotients show up in practice?