Spectroscopic studies of primitive asteroid families provide constraints on the composition of the solar nebula and the distribution of volatiles in the asteroid belt. Results from visible and near-infrared spectroscopy show diversity between primitive families. We aim 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 11 Themis asteroids and nine Veritas asteroids, for a total of 20 asteroids. We report the presence of a broad 10-μm emission feature, attributed to a layer of fine-grained silicates, in the spectra of all 11 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 low to moderate albedos. 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 are consistent with icy bodies with some amount of fine-grained silicates, but with coarser grains or denser surface structure than Trojan asteroids and comets. The range of spectral contrast of the 10-μm emission feature within each family suggests diversity in regolith porosity and/or grain size.