We study a combined sample of 264 star-forming, 51 composite, and 73 active galaxies using optical spectra from SDSS and mid-infrared (mid-IR) spectra from the Spitzer Infrared Spectrograph. We examine optical and mid-IR spectroscopic diagnostics that probe the amount of star formation and relative energetic contributions from star formation and an active galactic nucleus (AGN). Overall we find good agreement between optical and mid-IR diagnostics. Misclassifications of galaxies based on the SDSS spectra are rare despite the presence of dust obscuration. The luminosity of the [NeII] 12.8 \mu m emission-line is well correlated with the star formation rate (SFR) measured from the SDSS spectra, and this holds for the star forming, composite, and AGN-dominated systems. AGN show a clear excess of [NeIII] 15.6 \mu m emission relative to star forming and composite systems. We find good qualitative agreement between various parameters that probe the relative contributions of the AGN and star formation, including: the mid-IR spectral slope, the ratio of the [NeV] 14.3 \mu m to [NeII] \mu m 12.8 fluxes, the equivalent widths of the 7.7, 11.3, and 17 $\mu m$ PAH features, and the optical "D" parameter which measures the distance a source lies from the locus of star forming galaxies in the optical BPT emission-line diagnostic diagram. We also consider the behavior of the three individual PAH features by examining how their flux ratios depend upon the degree of AGN-dominance. We find that the PAH 11.3 \mu m feature is significantly suppressed in the most AGN-dominated systems.