Spectroscopic investigations of primitive asteroid families constrain family evolution and composition and conditions in the solar nebula, and reveal information about past and present distributions of volatiles in the solar system. Visible and near-infrared studies of primitive asteroid families have shown spectral diversity between and within families. Here, we aim to better understand the composition and physical properties of two primitive families with vastly different ages: ancient Themis (∼2.5 Gyr) and young Veritas (∼8 Myr). We analyzed the 5 - 14 μm Spitzer Space Telescope spectra of 11 Themis-family asteroids, including eight previously studied by Licandro et al. (2012), and nine Veritas-family asteroids, for a total of 20 asteroids in our sample. We detect a broad 10-μm emission feature, attributed to fine-grained and/or porous silicate regolith, in all 11 Themis-family spectra and six of nine Veritas-family asteroids, with 10-μm spectral contrast ranging from 1% ± 0.1% to 8.5% ± 0.9%. We used thermal modeling to derive diameters, beaming parameters and albedos for our sample. Asteroids in both families have beaming parameters near unity and geometric albedos in the range 0.03 - 0.14. Spectral contrast of the 10-μm silicate emission feature is correlated with beaming parameter and rotation period in the Themis family, and may be related to near-infrared spectral slope for both families. We see no correlations of 10-μm emission with diameter or albedo for either family. Comparison with laboratory spectra of primitive meteorites suggests these asteroids are similar to meteorites with relatively low abundances of phyllosilicates. Overall, our results suggest the Themis and Veritas families are primitive asteroids with variation in composition and/or regolith properties within both families.