We compare 13 near-infrared (0.8-2.4 μm) spectra of two low albedo C complex outer-belt asteroid families: Themis and Veritas. The disruption ages of these two families lie at opposite extremes: 2.5 ± 1.0 Gyr and 8.7 ± 1.7 Myr, respectively. We found striking differences between the two families, which show a range of spectral shapes and slopes. The seven Themis family members (older surfaces) have "red" (positive) slopes in the 1.6-2.4 μm region; in contrast, the six Veritas members (younger surfaces) have significantly "flatter" slopes at these same wavelengths. Moreover, the two families are characterized by different concavity at shorter (1.0-1.5 μm) wavelengths with the Themis group being consistently flat or concave up (smile) and the Veritas group being consistently concave down (frown). Each family contains a broad range of diameters, suggesting our results are not due to comparisons of asteroids of different sizes. The statistically significant clustering of the two spectral groups could be explained by one of the following three possibilities or a combination of them: (1) space weathering effects, (2) differences in original composition, or (3) differences in thermal history perhaps as a result of the difference in parent body sizes. As a result of our analyses, we propose a new method to quantify broad and shallow structures in the spectra of primitive asteroids. We found reasonable matches between the observed asteroids and individual carbonaceous chondrite meteorites. Because these meteoritic fits represent fresh surfaces, space weathering is neither necessary nor ruled out as an explanation of spectral differences between families. The six Veritas family near-infrared (NIR) spectra represent the first NIR analysis of this family, thus significantly increasing our understanding of this family over these wavelengths.