  We introduce a metric on Hilbert modules equipped with a generalized form of
a differential structure, thus extending Gromov-Hausdorff convergence theory to
vector bundles and quantum vector bundles --- not convergence as total space
but indeed as quantum vector bundle. Our metric is new even in the classical
picture, and creates a framework for the study of the moduli spaces of modules
over C*-algebras from a metric perspective. We apply our construction, in
particular, to the continuity of Heisenberg modules over quantum $2$-tori.
