  We compare molecular gas traced by 12CO(2-1) maps from the HERACLES survey,
with tracers of the recent star formation rate (SFR) across 30 nearby disk
galaxies. We demonstrate a first-order linear correspondence between Sig_mol
and Sig_SFR but also find important second-order systematic variations in the
apparent molecular gas depletion time, t_dep^mol = Sig_mol / Sig_SFR. At our 1
kpc common resolution, CO correlates closely with many tracers of the recent
SFR. Weighting each line of sight equally and using a fixed, Milky Way
alpha_CO, our data yield a molecular gas depletion time,
t_dep^mol=Sig_mol/Sig_SFR ~ 2.2 Gyr with 0.3 dex scatter, in good agreement
with literature data. We apply a forward-modeling approach to constrain the
power-law index, N, that relates the SFR surface density and the molecular gas
surface density and find N=1+/-0.15 for our full data set with some variation
from galaxy to galaxy. However, we caution that a power law treatment
oversimplifies the topic given that we observe correlations between t_dep^mol
and other local and global quantities. The strongest of these are a decreased
t_dep^mol in low-mass, low-metallicity galaxies and a correlation of the
kpc-scale t_dep^mol with dust-to-gas ratio, D/G. These correlations can be
explained by a CO-to-H2 conversion factor that depends on D/G in the
theoretically expected way. This is not a unique interpretation, but external
evidence of conversion factor variations makes it a conservative one. After
applying a D/G-dependent alpha_CO, some weak correlations between t_dep^mol and
local conditions persist. In particular, we observe lower t_dep^mol and
enhanced CO excitation associated with some nuclear gas concentrations. These
appear to reflect real enhancements in the SFR/H2 and t_dep appears multivalued
at fixed Sig_mol, supporting the the idea of "disk" and "starburst" modes
driven by environmental factors.
