I learned the definition of a sheaf from Hartshorne---that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. While fairly abstract at the outset, this seems to be (to me) an intuitive view; in particular, all of the manipulations and constructions with sheaves fit nicely into this schema.
I know that the older view of a sheaf on $X$ was to consider it as a triple $$ (E, X, \pi) $$ where $\pi : E \to X$ is a local homeomorphism, and so that the "sheaf of sections" of this map $\pi$ is the sheaf in the functorial sense described above.
This view makes much less sense to me, but I have to wonder if that is simply due to my having learned it second. However, it also makes me wonder if I am missing something, and so my question is as follows.
What are some (edit:) specific benefits of viewing a sheaf in this sense? What is gained by considering a sheaf as the espace étalé over $X$?