As Geoff explains, modular representations of finite groups do play an important role in some of the standard group-theoretic developments (although they play less of a role in the classification of simple groups than some people had expected initially).  In fact, I think Richard Brauer began to develop these ideas, including block theory, in the 1940s because he saw connections with structural questions about finite groups.   By now of course the subject is also pursued just because of its intrinsic beauty, but it has applications.  (As Dylan Wilson points out in his comment, some aspects of the modular theory played a key role for Quillen in his proof of the Adams conjecture, and these are emphasized in Serre's lecture notes.)

I'd add that for finite groups of Lie type defined over fields of prime characteristic $p$, the modular theory relative to $p$ has been helpful at times in the study of ordinary (characteristic 0) representations of those groups.   Moreover, it has implications for the rational representations of ambient algebraic groups, due to the close connections between irreducible representations of the finite groups and algebraic groups (Curtis, Steinberg, ...)

In a different direction, the study of Galois representations (and $p$-adic representations in general) has often required information about the modular representations of the finite groups of Lie type which occur naturally as quotients when reduction mod $p$ is applied to rings of $p$-adic integers.  All of this is complicated to explain, but is part of the broader study of certain matrix groups over local fields and their representations.   Which in turn has implications for the unifying program laid out by Langlands.