How useful are geometric aspects when studying finite groups?

It depends what you mean by "geometric aspects". I think the place that your question comes from is that nobody uses the methods commonly referred to as "Geometric Group Theory" (such as those explained, for example, here [PDF of A Course in Geometric Group Theory, by Brian Bowditch), then the answer is on page 24: many of the main tools of Geometric Group Theory involve studying things up to quasi-isomorphism, and all finite groups are quasi-isomorphic to each other, so there isn't much to do.

As a secondary thing, the reason for your impression might be rather the opposite: it's not that geometric methods are less common in finite group theory, it's that other methods are more common. That is: a lot of the geometric stuff with infinite groups still works fine for finite groups, it's just rather drowned out by the significantly larger body of other tools that we can use.

However, there are very rich areas of group theory which do use geometric methods, interpreted more broadly. A handful have been mentioned in the comments, but I'll go for something else (mostly because it's what I know about):

Given a (finite) group $G$, if we want to study its representations, we can often have a problem of getting our hands on those representations in a form that's amenable to further study (pick any finite group whose irreducible representations are not thoroughly known, and this is likely at play there somewhere). So it can be useful to have ways to generate representations of a given group, which you can then work with.

One reasonably common way to do this is to take an action of $G$ on something that has a homology theory (for our purposes, some geometric structure $X$), compute the homology of $X$ (in the normal topological sense, with some suitable coefficients), and pass the action of $G$ on $X$ through to get an action of $G$ on the homology terms, which are modules, and therefore give you representations of $G$. As a few examples: you can take the building of $G$, which gives some reasonably well-understood representations, you can take the simplicial complex of the poset of the subgroups of $G$, which gives you some somewhat less well-understood groups (but give me a bit on that, I'm working on it!), or you can take some complex derived from some natural action of your group (for example, the action of $S_n$ on a set of $n$ points gives an action on the set of collections of (proper) disjoint subsets of $S_n$, which form a simplicial complex with the subsets as vertices in the obvious way, which gives you some nice representations), or you can take the subgroup complexes mentioned in the comments.