No branches of mathematics are comparable, but for some question, I would consider Number Theory and Group Theory. These are (known to me) the branches of mathematics, which much much vastly developed in various directions due to a single problem:
Fermat's last theorem: it has much influence on development of Algebraic number theory and various other branches of number theory.
Classification of finite simple groups (posed by Holder): it influenced the development of representation theory, through which, various branches of Group Theory get developed (solvable groups, nilpotent groups).
Now coming to a question: in number theory, there are always emerging new branches in connection with F.L.T, and the contribution of any young mathematician in one branch always gets used, in some way, in the development of other branches. Thus, the branches of number theory which emerged from F.L.T are continuously growing.
In comparison to development of number theory by F.L.T., what is the effect of classification of finite simple groups in development of group theory (after the classification completed)?