  Based on relative energy estimates, we study the stability of solutions to
the Cahn-Hilliard equation with concentration dependent mobility with respect
to perturbations. As a by-product of our analysis, we obtain a weak-strong
uniqueness principle on the continuous level under realistic regularity
assumptions on strong solutions. We then show that the stability estimates can
be further inherited almost verbatim by appropriate Galerkin approximations in
space and time. This allows us to derive sharp bounds for the discretization
error in terms of certain projection errors and to establish order-optimal
a-priori error estimates for semi- and fully discrete approximation schemes.
