NGC 1097 is a nearby Seyfert 1 galaxy with a bright circumnuclear starburst ring, a strong large-scale bar, and an active nucleus. We present a detailed study of the spatial variation of the far-infrared (FIR) CII 158 mu m and OI63 mu m lines and mid-infrared H sub(2) emission lines as tracers of gas cooling, and of the polycyclic aromatic hydrocarbon (PAH) bands as tracers of the photoelectric heating, using Herschel-PACS and Spitzer-IRS infrared spectral maps. We focus on the nucleus and the ring, and two star-forming regions (Enuc N and Enuc S). We estimated a photoelectric gas heating efficiency (CII158 mu +OI63 mu m)/PAH in the ring about 50% lower than in Enuc N and S. The average 11.3/7.7 mu m PAH ratio is also lower in the ring, which may suggest a larger fraction of ionized PAHs, but no clear correlation with CII158 mu m/PAH(5.5-14 mu m) is found. PAHs in the ring are responsible for a factor of two more CII158 mu m and OI63 mu m emission per unit mass than PAHs in the Enuc S. spectral energy distribution (SED) modeling indicates that at most 25% of the FIR power in the ring and Enuc S can come from high-intensity photodissociation regions (PDRs), in which case G sub(0) ~ 10 super(2.3) and n sub(H) ~ 10 super(3.5) cm super(-3) in the ring. For these values of G sub(0) and n sub(H), PDR models cannot reproduce the observed H sub(2) emission. Much of the H sub(2) emission in the starburst ring could come from warm regions in the diffuse interstellar medium that are heated by turbulent dissipation or shocks.