Using infrared imaging from the Herschel Space Observatory, observed as part of the Very Nearby Galaxies Survey, we investigate the spatially resolved dust properties of the interacting Whirlpool galaxy system (NGC 5194 and NGC 5195), on physical scales of similar to 1 kpc. Spectral energy distribution modeling of the new infrared images in combination with archival optical and near-through mid-infrared images confirms that both galaxies underwent a burst of star formation similar to 370-480 Myr ago and provides spatially resolved maps of the stellar and dust mass surface densities. The resulting average dust-to-stellar mass ratios are comparable to other spiral and spheroidal galaxies studied with Herschel, with NGC 5194 at log(M-dust/M-star) = -2.5 +/- 0.2 and NGC 5195 at log(M-dust/M-star) = -3.5 +/- 0.3. The dust-to-stellar mass ratio is constant across NGC 5194 suggesting the stellar and dust components are coupled. In contrast, the mass ratio increases with radius in NGC 5195 with decreasing stellar mass density. Archival mass surface density maps of the neutral and molecular hydrogen gas are also folded into our analysis, revealing a fairly constant gas-to-dust mass ratio, 94 +/- 17 across the system. Somewhat surprisingly, we find the dust in NGC 5195 is heated by a strong interstellar radiation field (ISRF), over 20 times that of the ISRF in the Milky Way, resulting in relatively high characteristic dust temperatures (similar to 30 K). This post-starburst galaxy contains a substantial amount of low-density molecular gas and displays a gas-to-dust ratio (73 +/- 35) similar to spiral galaxies. It is unclear why the dust in NGC 5195 is heated to such high temperatures as there is no star formation in the galaxy and its active galactic nucleus is 5-10 times less luminous than the one in NGC 5194, which exhibits only a modest enhancement in the amplitude of its ISRF.