  The Gromov--Hausdorff distance measures the difference in shape between
compact metric spaces. While even approximating the distance up to any
practical factor poses an NP-hard problem, its relaxations have proven useful
for the problems in geometric data analysis, including on point clouds,
manifolds, and graphs. We investigate the modified Gromov--Hausdorff distance,
a relaxation of the standard distance that retains many of its theoretical
properties, which includes their topological equivalence on a rich set of
families of metric spaces. We show that the two distances are
Lipschitz-equivalent on any family of metric spaces of uniformly bounded size,
but that the equivalence does not hold in general, not even when the distances
are restricted to ultrametric spaces. We additionally prove that the standard
and the modified Gromov--Hausdorff distances are either equal or within a
factor of 2 from each other when taken to a regular simplex, which connects the
relaxation to some well-known problems in discrete geometry.
