We use mid-infrared (MIR) spectroscopy from the Spitzer Infrared Spectrograph to study the nature of star-formation and supermassive black hole accretion for a sample of 65 IR-luminous galaxies at 0.02 < z < 0.6 with F(24 μm) > 1.2 mJy. The MIR spectra cover wavelengths 5-38 μm, spanning the polycyclic aromatic hydrocarbon (PAH) features and important atomic diagnostic lines. Our sample of galaxies corresponds to a range of total IR luminosity, L IR = L(8-1000 μm) = 1010-1012 L ☉ (median L IR of 3.0 × 1011 L ☉). We divide our sample into a subsample of galaxies with Spitzer Infrared Array Camera 3.6-8.0 μm colors indicative of warm dust heated by an active galactic nucleus (AGN; IRAGN) and those galaxies whose colors indicate star-formation processes (non-IRAGN). Compared to the non-IRAGN, the IRAGN show smaller PAH emission equivalent widths, which we attribute to an increase in mid-IR continuum from the AGN. We find that in both the IRAGN and star-forming samples, the luminosity in the PAH features correlates strongly with [Ne II] λ12.8 μm emission line, from which we conclude that the PAH luminosity directly traces the instantaneous star-formation rate (SFR) in both the IRAGN and star-forming galaxies. We compare the ratio of PAH luminosity to the total IR luminosity, and we show that for most IRAGN star-formation accounts for 10%-50% of the total IR luminosity. We also find no measurable difference between the PAH luminosity ratios of L 11.3/L 7.7 and L 6.2/L 7.7 for the IRAGN and non-IRAGN, suggesting that AGN do not significantly excite or destroy PAH molecules on galaxy-wide scales. Interestingly, a small subset of galaxies (8 of 65 galaxies) show a strong excess of [O IV] λ25.9 μm emission compared to their PAH emission, which indicates the presence of heavily-obscured AGN, including 3 galaxies that are not otherwise selected as IRAGN. The low PAH emission and low [Ne II] emission of the IRAGN and [O IV]-excess objects imply the IR luminosity of these objects is dominated by processes associated with the AGN. Because these galaxies lie in the "green valley" of the optical color-magnitude relation and have low implied SFRs, we argue their hosts have declining SFRs and these objects will transition to the red sequence unless some process restarts their star-formation.