Your final question, i.e. in what way does one compare the classical and the étale topology of a scheme, is answered for instance in section 2 of Mumford's classical "Picard Groups of Moduli Problems". There are also some nicely written parts in Vistoli's notes on descent on how to compare Grothendieck topologies in general.
As you say there is no direct way to compare the two. What you do is introduce an auxilliary topology which refines both of them. If you take a sep. finite type scheme X over $\mathbf C$, then you can put a topology on complex analytic spaces over X by taking for open subsets those maps $U \to X(\mathbf C)$ that form a covering space over an open subset of $X(\mathbf C)$. Coverings are jointly surjective. We call this site $X_{cx}^\ast$. Then every every open set in both the étale and classical topologies are also open sets of this site, so there are maps $\alpha : X_{cx}^\ast \to X_{cx}$ and $\beta : X_{cx}^\ast \to X_{ét}$. Moreover, $\alpha$ is an equivalence of topologies. (I see now that most of what I said here was already stated in the comment by Ryan Reich.)
