  Multi-wavelength time-resolved observations of rotationally modulated
variability from brown dwarfs and giant exoplanets are the most effective
method for constraining their heterogeneous atmospheric structures. In a
companion paper (Bowler et al. 2020), we reported the discovery of strong
near-infrared variability in HST/WFC3/G141 light curves of the very red L-dwarf
companion VHS J1256-1257b. In this paper, we present a follow-up 36-hr
Spitzer/IRAC Channel 2 light curve together with an in-depth analysis of the
HST and the Spitzer data. The combined dataset provides time-resolved light
curves of VHS1256b sampling 1.1 to 4.5 $\mu$m. The Spitzer light curve is
best-fit with a single sine wave with a period of $22.04\pm0.05$ hr and a
peak-to-peak amplitude of $5.76\pm0.04$%. Combining the period with a
previously measured projected rotational velocity ($v\sin i$), we find that
VHS1256b is most consistent with equatorial viewing geometry. The
HST/G141+Spitzer spectral energy distribution favors a $T_{\rm eff}$ of 1000~K,
low surface gravity model with disequilibrium chemistry. The spectral
variability of VHS1256b is consistent with predictions from partly cloudy
models, suggesting heterogeneous clouds are the dominant source of the observed
modulations. We find evidence at the $3.3\sigma$ level for amplitude variations
within the 1.67$\mu$m CH$_{4}$ band, which is the first such detection for a
variable L-dwarf. We compare the HST/G141 time-resolved spectra of three red
L-dwarfs with high-amplitude near-infrared rotational modulations and find that
although their time-averaged spectra are similar, their spectroscopic
variabilities exhibit notable differences. This diversity reinforces the
advantage of time-resolved spectroscopic observations for understanding the
atmospheres of brown dwarfs and directly-imaged exoplanets.
