We combine new ASTE/AzTEC 1.1 mm maps of the galaxy pair NGC 1512/1510 with archival Spitzer IRAC and MIPS images covering the wavelength range 3.6-160 μm from the SINGS project. The availability of the 1.1 mm map enables us to measure the long-wavelength tail of the dust emission in each galaxy, and in sub-galactic regions in NGC 1512, and to derive accurate dust masses. The two galaxies form a pair consisting of a large, high-metallicity spiral (NGC 1512) and a low-metallicity, blue compact dwarf (NGC 1510), which we use to compare similarities and contrast differences. Using the models of Draine and Li, the derived total dust masses are (2.4 ± 0.6) × 107 M and (1.7 ± 3.6) × 105 M for NGC 1512 and NGC 1510, respectively. The derived ratio of dust mass to H I gas mass for the galaxy pair, 0.0034, is much lower (by at least a factor of 3) than expected, as previously found by Draine et al. In contrast, regions within NGC 1512, specifically the central region and the arms, do not show such unusually low ratios; furthermore, the dust-to-gas ratio is within expectations for NGC 1510. These results suggest that a fraction of the H I included in the determination of the ratio of the NGC 1512/NGC 1510 pair is not associated with the star-forming disks/regions of either galaxy. Using the dust masses derived from the models of Draine & Li as references, we perform simple two-temperature modified blackbody fits to the far-infrared/millimeter data of the two galaxies and the sub-regions of NGC 1512, in order to derive and compare the dust masses associated with warm and cool dust temperature components. As generally expected, the warm dust temperature of the low-metallicity, low-mass NGC 1510 (Tw ~ 36 K) is substantially higher than the corresponding warm temperature of the high-metallicity spiral NGC 1512 (Tw ~ 24 K). In both galaxies (albeit with a large uncertainty for NGC 1510), our fits indicate that a substantial fraction (>93%) of the total dust mass is in a cool dust component, with temperatures ~14-16 K for NGC 1512 and ~15-24 K for NGC 1510. This result is similar to what is determined for a few other nearby galaxies. In contrast, the warm dust component in the sub-galactic regions of NGC 1512 represents a much larger fraction of the total dust content, in agreement with the fact that all three regions have higher specific star formation rates than the average in the galaxy; in the center, the warm dust represents about 40% of the total, while in the arms the fractions are close to ~20%.