I hope this is not too vague of a question. Stone duality implies that the category Pro(FinSet) is equivalent to the category of Stone spaces (compact, Hausdorff, totally disconnected, topological spaces). This equivalence carries over to profinite vs Stone-topological algebras for a number of algebraic theories, such as groups, monoids, semigroups, and rings. The case of profinite groups is especially well-known.
My question is: why are such equivalences important? Where in mathematics do we gain something by identifying a pro-(finite group) with a Stone topological group? I mean something other than "concreteness" or "familiarity"—certainly it may be easier (for some people) to think about a Stone topological group than about a cofiltered diagram of finite groups, but are there important things that we couldn't prove about cofiltered diagrams of finite groups without knowing that they are equivalent to Stone topological groups?
I am especially interested because this manifestation of Stone duality seems to be "fragile" for generalizations in several directions. For instance, Theo JF commented on this question that Stone-topological groupoids are not equivalent to pro-(finite groupoids). The equivalence is also false if we generalize from finite sets/groups to ones of larger cardinality. It is true that pro-groups with surjective transition maps can be identified with pro-discrete locales, but I don't know anything about whether this is true for pro-sets (cf. question linked above), or any type of algebras other than groups. So in all the cases where the generalization fails, what is lost if we just work with pro-objects and ignore the missing topological aspect?
Best Answer
The answer is "for lots of reasons". Let me explain a couple:
1. As several people have already noted in answers and comments, sometimes profinite groups arise naturally. For example, if $k$ is a field and $k^s$ is its separable closure, it is natural to consider the group of automorphisms of $k^s$ over $k$. One then equips this group with a topology (the weak topology, when $k^s$ is equipped with its discrete topology --- i.e. two elmenents are close it their action coincides on a large finite set of elements of $k^s$), and then discovers that this makes the automorphism group into a profinite group.
Now one can bring the pro-structure to the fore by regarding $k^s$ not just as a field extension of $k$, but as an ind-finite extension of $k$, by writing it as the inductive limit of its finite subextensions. But, while this is technically useful in some contexts (for example, in the proof the automorphism group is profinite), it is not always convenient --- there are often advantages to having $k^s$ available as a naked field, without having to bother with its ind-structure.
2. The concept of topology is incredibly, amazingly flexible, much more so than the concept of pro-object. There are lots of illustrations of this, but one very convincing one is the theory of the adeles. Here one takes the topological product of a profinite ring with copies of $\mathbb R$ and $\mathbb C$. One obtains a locally compact ring, equips it with a Haar measure, and proceeds to do harmonic analysis. Trying to carry all this out in the language of pro-systems (say, of pro-Lie groups) would be incredibly convoluted. Indeed, in the early days of class field theory, before the introduction of the adelic view-point, this is essentially what people did: they worked explicitly with the pro-systems underlying the adeles (without using that language, of course). The introduction of ideles and adeles swept away the inherent (conceptual and notational) complexities of that view-point, and so was (and is) rightly regarded as a major advance.