I was going to write a comment, but I think I will make it an answer, which is far from comprehensive. 

   If one is interested in applications to the structure of finite groups, possibly the best-known example is the proof of Glauberman's Z*-theorem, which is an essential tool for the classification of finite simple groups ( I believe it was the most cited paper in Group Theory in the 1970's).
Modular representation theory was an indispensable tool in its proof, as in the proof of the Brauer-Suzuki theorem (concerning finite groups with (generalized) quaternion Sylow 2-subgroups) before it ( though Glauberman later published a rather longer proof of the Brauer-Suzuki theorem which only used ordinary character theory).
        In general, the uses of modular representation theory which contributed towards the Classification of Finite Simple groups tended to be in groups whose Sylow $2$-subgroups did not contained large elementary Abelian Sylow $2$-subgroups ( ``large" usually meaning of order at least 8). So classification of finite groups with generalized quaternion, dihedral, or semi-dihedral, Sylow $2$-subgroups make use of results from block theory, though in some cases, ordinary character-theoretic proofs eventually replaced the uses of block theory.
    Othr ways in which modular representation theory has played a role in finite group theory include teh Hall-Higman theorem, which certainly influenced J.G. Thompson's early work, including his Ph.D. thesis.
 However, restricting attention to immediate applications to finite group theory is presenting an unrepresentative picture. Other areas where modular representations of finite groups play a role include Number Theory ( eg they arise in Wiles' proof of Fermat's Last Theorem), and areas such as Coding Theory, among many others.