In what sense is the étale topology equivalent to the Euclidean topology?

Saying that the étale topology is equivalent to the euclidean topology is vastly overstating the case. For example, if you compute the cohomology of a complex algebraic variety with coefficients in $\mathbb Q$ in the étale topology, typically you get 0. On the other hand, it is a deep result that the étale cohomology of such a variety with coefficients in a finite abelian group coincides with its cohomology in the euclidean topology.

Similarly, you can't capture the whole fundamental group with the étale topology, but only its finite quotients (and the fact that you can indeed describe the finite quotients of the fundamental group via étale covers is, again, a deep result).

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.)

Here is another take on your question, in the direction of (ramified) covers.
1. Riemann There is an analytification functor $X\mapsto X^{an}$ from the category of $\mathbb C$-schemes locally of finite type to that of (non-reduced) complex analytic spaces. Its introduction is due principally to Riemann, Chow, Serre and Grothendieck. It has all the desirable properties and of course the set of points of $X^{an}$ is $X(\mathbb C)$. Riemann's existence theorem states that this functor induces an equivalence of categories between the finite étale covers of $X$ and the finite étale analytic covers of $X^{an}$.
This is the deep result (alluded to in Angelo's fine answer) which, in particular, yields the identification of the topological fundamental group of $X^{an}$ with a completion of the scheme-theoretic fundamental group.

2. Grauert-Remmert In a sense the classification of algebraic covers has been reduced to that of analytic ones. We can then apply the following result, due to Grauert and Remmert :
Let $X$ be a normal analytic space and $U\subset X$ an open subset with analytic complement. Then any (ramified) finite normal cover of $U$ uniquely extends to a normal finite cover of $X$.
(Grothendieck, in SGA 1, gave a slick proof of this theorem by invoking Hironaka's resolution of singularities)