  In this paper, we obtain a complete characterization for the compact
difference of two composition operators acting on Bergman spaces with weight
$\omega=e^{-\eta}$, $\Delta\eta>0$ in terms of the $\eta$-derived
pseudodistance of two analytic self maps. In addition, we provide simple
inducing maps which support our main result. We also study the topological path
component of the space of all bounded composition operators on $A^2(\omega)$
endowed with the Hilbert-Schmidt norm topology.
