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 https://mathoverflow.net/questions/17786/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.)