Background and motivation: I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch how to compute the "fundamental group of a field," primarily as a way of illustrating the analogy between Galois theory and covering space theory, but also because this is the sort of thing that makes me eager to learn more about a subject when I am on the receiving end of a lecture.
Unfortunately, as yet, I myself know little more about the etale fundamental group than the definition and a bit of motivation (although I am certainly planning to learn more). In particular, I realized that there is an obvious question I don't know how to answer. When defining the fundamental group in an algebraic setting, we define it as an inverse limit because there is, in general, no natural analogue of a universal cover. However, if we are looking at Spec k, there is an obvious candidate, namely the Spec of the algebraic closure of k.
Question: Let $k$ be a field, with algebraic closure $K$. How does the etale fundamental group of Spec $k$ compare to the automorphism group of $K$ over $k$? Assuming they are different, what (in very general terms) are the reasons for working with one rather than the other?