Compositional studies of primitive asteroid families provide constraints on the physical and chemical environment of the solar nebula and the evolution of the asteroid belt. Spectroscopic studies in the visible and near-infrared have shown spectral diversity between primitive families. Our goal is to better constrain the composition of two primitive families with very different ages: Themis (~2.5 Gyr) and Veritas (~8 Myr). We analyzed 5 – 14 μm Spitzer Space Telescope spectra of a total of 18 asteroids, nine from each family. We report the presence of a broad 10-μm emission feature, attributed to a layer of fine-grained silicates, in the spectra of all nine Themis asteroids and six of nine Veritas asteroids in our sample. Spectral contrast in statistically significant detections of the 10-μm feature ranges from 1% ± 0.1% to 8.5% ± 0.9%. Comparison with the spectra of primitive meteorites (McAdam et al. 2015, Icarus, 245, 320) suggests asteroids in both families are similar to meteorites with lower abundances of phyllosilicates. We used the Near-Earth Asteroid Thermal Model 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 0.06 ± 0.01 to 0.14 ± 0.02. We find that contrast of the silicate emission feature is not correlated with asteroid diameter; however, higher 10-μm contrast may be associated with flatter spectral slopes in the near-infrared. The spectra of both families suggest icy bodies with some amount of fine-grained silicates, but with coarser grains or denser surface structure than Trojan asteroids and comets.