We present a detailed spectroscopic and photometric analysis of DA and DB white dwarfs drawn from the Sloan Digital Sky Survey with trigonometric parallax measurements available from the Gaia mission. The temperature and mass scales obtained from fits to $ugriz$ photometry appear reasonable for both DA and DB stars, with almost identical mean masses of $langle M rangle = 0.617~M_odot$ and $0.620~M_odot$, respectively. The comparison with similar results obtained from spectroscopy reveals several problems with our model spectra for both pure hydrogen and pure helium compositions. In particular, we find that the spectroscopic temperatures of DA stars exceed the photometric values by $sim$10% above $T_{rm eff}sim14,000$~K, while for DB white dwarfs, we observe large differences between photometric and spectroscopic masses below $T_{rm eff}sim16,000$~K. We attribute these discrepancies to the inaccurate treatment of Stark and van der Waals broadening in our model spectra, respectively. Despite these problems, the mean masses derived from spectroscopy --- $langle M rangle = 0.615~M_odot$ and $0.625~M_odot$ for the DA and DB stars, respectively --- agree extremely well with those obtained from photometry. Our analysis also reveals the presence of several unresolved double degenerate binaries, including DA+DA, DB+DB, DA+DB, and even DA+DC systems. We finally take advantage of the Gaia parallaxes to test the theoretical mass-radius relation for white dwarfs. We find that 65% of the white dwarfs are consistent within the 1$sigma$ confidence level with the predictions of the mass-radius relation, thus providing strong support to the theory of stellar degeneracy.
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