In my own research I use profinite groups quite frequently (for Galois groups and etale fundamental groups). However my use of them amounts to book-keeping: I only care about finite levels (finite Galois extensions; finite covers) and so I take their inverse limit. Then various "topological" arguments just mean: look at the finite levels (this occurs because the quotients by the open subgroups exactly correspond to the finite levels I care about.)
There are people who use profinite groups more intently than I do, and I suspect that they see some value in them as mathematical objects. So my question is this:
Question
What theorems/properties are good about profinite groups that don't arise trivially from its book-keeping nature (so for example the profinite Sylow theorems are disqualified, because they arise trivially from the finite-level group theory). What, if anything, rewards us for dealing with this new type of object rather than with the finite levels? (again, except that it makes it easier to write down notes.)
Best Answer
More generally, you might ask "what is the point of constructing limits or colimits of diagrams of objects instead of working directly with the diagrams?" A generic answer is that a diagram of objects in a category describes a functor, and it is useful to know that that functor is representable. For example, if objects $X_1, X_2$ in a category have a product $X_1 \times X_2$, this is equivalent to the statement that the functor $\text{Hom}(-, X_1) \times \text{Hom}(-, X_2)$ is representable. So you now know that this functor takes colimits to limits, which is new information.
Another generic answer is the following. Any time you construct an object $X$ as a limit of a diagram of other objects $X_i$ in a category, you know what the maps into $X$ look like by definition (compatible systems of maps into the $X_i$). What you don't know is what the maps out of $X$ look like, and this is new information you get from the existence of $X$. For example, the limit of the empty diagram is the terminal object $1$, and while maps into $1$ are trivial, maps out of $1$ ("global points") are not; in the category of schemes over a field $k$, for example (an example within an example!), they correspond to $k$-points.
Specializing to Galois theory, when you construct a Galois group $G$ as a limit of finite Galois groups $G_i$, the new information you have access to is, for example, the representation theory of $G$. I don't know how one would talk about the correspondence between modular forms and 2-dimensional Galois representations without direct access to the group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, for example.