In symplectic linear algebra this is perhaps not completely clear. However, if one passes to symplectic geometry then the Lagrangean submanifolds indeed play a dominant role. This is in some sense Weinstein's Lagrangean creed: Every manifold $M$ is a Lagrangean manifold when viewed as zero section inside its cotangent bundle.
But also beyond this observation Lagrangean submanifolds show up in e.g. Hamilton-Jacobi theory, in completely integrable systems, and in quantization theory where they can be thought of semiclassical limit of quantum states (to some extend). Finally, in semiclassical analysis they turn out to be related to supports of pseudo-differential and Fourier integral operators.
Surprisingly, symplectic submanifolds turn out to be not that important. Instead, coisotropic submanifolds have an important role when it comes to phase space reduction (coisotropic reduction). Also in Poisson geometry, coisotropic submaifolds are in some sense the closest one can get to Lagrangean ones, a notion which no longer makes sense.
I hope this gives some inspiration why Lagrangean submanifolds are useful.