TITLE: Why are profinite topologies important?
QUESTION [34 upvotes]: 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?

REPLY [5 votes]: Compact semigroups have nice properties (existence of idempotents, minimal ideals, etc). If you view a profinite semigroup as a compact semigroup you get all this for free. I am not even sure how to think of these things for a pro-object in the category of finite semigroups. 
Also one wants to think about free profinite groups and semigroups as missing free objects from the categories of finite groups and semigroups. That is the categories of profinite groups and semigroups have natural forgetful functors to sets which have left adjoints. It is not so natural from the pro-object viewpoint.