  We examine Euclidean plane domains with their hyperbolic or quasihyperbolic
distance. We prove that the associated metric spaces are quasisymmetrically
equivalent if and only if they are bi-Lipschitz equivalent. On the other hand,
for Gromov hyperbolic domains, the two corresponding Gromov boundaries are
always quasisymmetrically equivalent. Surprisingly, for any finitely connected
hyperbolic domain, these two metric spaces are always quasiisometrically
equivalent. We construct an example where the spaces are not quasiisometrically
equivalent.
