  Let $\varphi$ be a self-map of the unit disk and let $C_\varphi$ denote the
composition operator acting on the standard Dirichlet space $\mathcal{D}$. A
necessary condition for compactness of a difference of two bounded composition
operators acting on $\mathcal{D}$, is given. As an application, a
characterization of disk automorphisms $\varphi$ and $\psi$ for which the
commutator $[C^*_\psi,C_\varphi]$ is compact, is given.
