Big etale topos vs small etale topos Are they equivalent?
That is, given a sheaf of sets $\mathscr{F}$ defined on the small etale site on $X$, is there an essentially unique way to extend it to a sheaf on the big etale site on $X$? If not, what is an example of a sheaf which cannot be extended?
What about for sheaves of abelian groups?
 A: There is a site morphism
$$i: X_{\rm \acute{e}t}\to {\rm \acute{E}t}(X),$$
giving an adjunction (indeed, a geometric morphism of topoi—abstract nonsense)
$${\rm EXT}=i^*: \mathsf{Shv}(X_{\rm \acute{e}t})\rightleftarrows\mathsf{Shv}({\rm \acute{E}t}(X)): i_*={\rm Res},$$
where ${\rm EXT}=i^*$ is given by some Kan extension, $i_*={\rm Res}$ is really the restriction.
Standard text books on étale cohomology say that the unit $F\to{\rm Res}({\rm EXT}(F))$ of the adjunction $({\rm EXT}, {\rm Res})$ is an isomorphism.
So every sheaf on the small étale site is always extendable to the big étale site and restricts back to the original sheaf. So your question is not a correct one.
On the other hand, the counit ${\rm EXT}({\rm Res}(G))\to G$ is usually not  an isomorphism. As in the above comments, different big étale sheaves can restrict to isomorphic small étale sheaves. The correct question is to ask for such examples, already given by the above comments.
What is important in étale cohomology theory is, for any $A\in\mathsf{ShvAb}(X_{\rm \acute{e}t}), B\in\mathsf{ShvAb}({\rm \acute{E}t}(X)), U\in X_{\rm \acute{e}t}$, there are canonical isomorphisms
$${\rm H}^n_{\rm \acute{e}t}(X; A)\cong{\rm H}^n_{{\rm \acute{E}t}(X)}(X; {\rm EXT}(A)), {\rm H}^n_{\rm \acute{e}t}(X; {\rm Res}(B))\cong{\rm H}_{{\rm \acute{E}t}(X)}^n(X; B); $$
$${\rm H}^n_{\rm \acute{e}t}(U; A)\cong{\rm H}^n_{{\rm \acute{E}t}(X)}(U; {\rm EXT}(A)).$$
