We examine the relation between polycyclic aromatic hydrocarbon (PAH) emission at 8 mu m and far-infrared emission from hot dust grains at 24 mu m and from large dust grains at 160 and 250 mu m in the nearby spiral galaxies NGC 2403 and M83 using data from the Spitzer Space Telescope and Herschel Space Observatory. We find that the PAH emission in NGC 2403 is better correlated with emission at 250 mu m from dust heated by the diffuse interstellar radiation field (ISRF) and that the 8/250-mu m surface brightness ratio is well correlated with the stellar surface brightness as measured at 3.6 mu m. This implies that the PAHs in NGC 2403 are intermixed with cold large dust grains in the diffuse interstellar medium (ISM) and that the PAHs are excited by the diffuse ISRF. In M83, the PAH emission appears more strongly correlated with 160 mu m emission originating from large dust grains heated by star-forming regions. However, the PAH emission in M83 is low where the 24-mu m emission peaks within star-forming regions, and enhancements in the 8/160-mu m surface brightness ratios appear offset relative to the dust and the star-forming regions within the spiral arms. This suggests that the PAHs observed in the 8 mu m band are not excited locally within star-forming regions but either by light escaping non-axisymmetrically from star-forming regions or locally by young, non-photoionizing stars that have migrated downstream from the spiral density waves. The results from just these two galaxies show that PAHs may be excited by different stellar populations in different spiral galaxies.