Zariski cohomology of an etale sheaf vs etale cohomology.

Question

Let $\mathcal{F}_{Zar}$ be a Zariski sheaf (on the big site of schemes or $S$-schemes) and $\mathcal{F}_{et}$ be its etale sheafification on the big site. Let $X$ be a scheme. If we know that $H_{Zar}^i(X,\mathcal{F}_{Zar})\simeq H_{et}^i(X,\mathcal{F}_{et})$ (sheaf cohomology is taken wrt the small site of $X$) for all $i$ and $X$, can we say anything about the Zariski sheaf cohomology of $\mathcal{F}_{et}$? ($H_{Zar}^i(X,\mathcal{F}_{et})$)