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.
