  New developments in scanning tunneling spectroscopy now allow for the
spatially resolved measurement of the Josephson critical current $I_c$ between
a tip and a superconducting sample, a nearly direct measurement of the true
superconducting order parameter. However, it is unclear how these $I_c$
measurements are correlated with previous estimates of the spectral gap taken
from differential conductance measurements. In particular, recent such
experiments on an iron-based superconductor found almost no correlation between
$I_c$ and the spectral gap obtained from differential conductance $g=dI/dV$
spectra, reporting instead a more significant correlation between $I_c$ and the
the coherence-peak height. Here we point out that the correlation--or the lack
thereof--between these various quantities can be naturally explained by the
effect of disorder on unconventional superconductivity. Using large scale
numerical simulations of a BCS $d$-wave pair Hamiltonian with many-impurity
potentials, we observe that "substitutional" disorder models with weak
pointlike impurities lead to a situation in which the true superconducting
order parameter and $I_c$ are both uncorrelated with the spectral gap from
$dI/dV$ measurements and highly correlated with the coherence-peak heights. The
underlying mechanism appears to be the disorder-induced transfer of spectral
weight away from the coherence peaks. On the other hand, smooth impurity
potentials with a length scale larger than the lattice constant lead to a large
positive correlation between the true superconducting order parameter and the
spectral gap, in addition to a large correlation between the order parameter
and the coherence-peak height. We discuss the applicability of our results to
recent Josephson scanning tunneling spectroscopy experiments on iron-based and
cuprate high-temperature superconductors.
