  We present a comparative study of the observed properties of the optical and
X-ray afterglows of short- and long-duration gamma-ray bursts (GRBs). Using a
large sample of 37 short and 421 long GRBs, we find a strong correlation
between the afterglow brightness measured after 11 hours and the observed
fluence of the prompt emission. Both the optical (R band) and X-ray flux
densities (F_R and F_X) scale with the gamma-ray fluence, F_gamma. For bursts
with a known redshift, a tight correlation exists between the afterglow flux
densities at 11 hours (rest-frame) and the total isotropic gamma-ray energy,
Egi: F_{R,X} ~ Egi^{alpha}, with alpha ~ 1. The constant of proportionality is
nearly identical for long and short bursts, when Egi is obtained from the Swift
data. Additionally, we find that for short busts with F_gamma >10^{-7} erg
cm^{-2}, optical afterglows are nearly always detected by reasonably deep early
observations. Finally, we show that the ratio F_R/F_X has very similar values
for short and long bursts. These results are difficult to explain in the
framework of the standard scenario, since they require that either (1) the
number density of the surrounding medium of short bursts is typically
comparable to, or even larger than the number density of long bursts; (2) short
bursts explode into a density profile, n(r)\alpha r^{-2} or (3) the prompt
gamma-ray fluence depends on the density of the external medium. We therefore
find it likely that either basic assumptions on the properties of the
circumburst environment of short GRBs or else the standard models of GRB
emission must be re-examined. We believe that the most likely solution is that
the ambient density surrounding typical short bursts is higher than has
generally been expected: a typical value of ~1 per cm^{-3} is indicated.
