The short answer is that locally isomorphic things needn't be globally isomorphic, and this isn't specific to the etale topology. Let me spell it out for locally constant sheaves of vector spaces on an ordinary (sufficiently nice) topological space $X$. Such sheaves correspond to representations of the fundamental group (see Why are local systems and representations of the fundamental group equivalent). Two locally constant sheaves $F$ and $G$ of the same rank are locally isomorphic, and in fact they pullback to to isomorphic sheaves on the universal cover $\tilde X\to X$. However, they won't be isomorphic unless the corresponding representations match. This is entirely analagous to the example of the nonisomorphic sheaves $\mathbb{Z}/n\mathbb{Z}$ and $\mu_n$ pulling to isomorphic sheaves on $Spec( k^{sep})$.
(As I was writing this, I realize that Emerton has already given an answer, but perhaps two is better than none.)