  When $\varphi$ and $\psi$ are linear-fractional self-maps of the unit ball
$B_N$ in ${\mathbb C}^N$, $N\geq 1$, we show that the difference
$C_{\varphi}-C_{\psi}$ cannot be non-trivially compact on either the Hardy
space $H^2(B_N)$ or any weighted Bergman space $A^2_{\alpha}(B_N)$. Our
arguments emphasize geometrical properties of the inducing maps $\varphi$ and
$\psi$.
