I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures.  I'd like to know what other problems one can solve using the machinery of etale cohomology.  I know a little bit about how etale cohomology groups appear in algebraic number theory but I'd like to know about ways that these things show up in other mathematical subjects as well.  Is there anything that an algebraic topologist should really know about etale cohomology?  What about a differential geometer?  
$\DeclareMathOperator{\gal}{Gal}$
Here's a comment which one can make to differential geometers which at least explains what etale cohomology "does". Given an algebraic variety over the reals, say a smooth one, its complex points are a complex manifold but with a little extra structure: the complex points admit an automorphism coming from complex conjugation. Hence the singular cohomology groups inherit an induced automorphism, which is extra information that is sometimes worth carrying around. In short: the cohomology of an algebraic variety defined over the reals inherits an action of $\gal(\mathbb{C}/\mathbb{R})$.

The great thing about etale cohomology is that a number theorist can now do the same trick with algebraic varieties defined over $\mathbb{Q}$. The etale cohomology groups of this variety will have the same dimension as the singular cohomology groups (and are indeed isomorphic to them via a comparison theorem, once the coefficient ring is big enough) but the advantage is that that they inherit a structure of the amazingly rich and complicated group $\gal(\bar{\mathbb{Q}}/\mathbb{Q})$. I've often found that this comment sees off differential geometers, with the thought "well at least I sort-of know the point of it now". A differential geometer probably doesn't want to study $\gal(\bar{\mathbb{Q}}/\mathbb{Q})$ though.

If I put my Langlands-philosophy hat on though, I can see a huge motivation for etale cohomology: Langlands says that automorphic forms should give rise to representations of Galois groups, and etale cohomology is a very powerful machine for constructing representations of Galois groups, so that's why I might be interested in it even if I'm not an algebraic geometer.

Finally, I guess a much simpler motivating good reason for etale cohomology is that geometry is definitely facilitated when you have cohomology theories around. That much is clear. But if you're doing algebraic geometry over a field that isn't $\mathbb C$ or $\mathbb R$ then classical cohomology theories aren't going to cut it, and the Zariski topology is so awful that you can't use it alone to do geometry---you're going to need some help. Hence etale cohomology, which gives the right answers: e.g. a smooth projective curve over any field has a genus, and etale cohomology is a theory which assigns to it an $H^1$ of dimension $2g$ (<pedant> at least if you use $\ell$-adic cohomology for $\ell$ not zero in the field <\pedant>).
