Closely spaced differential cross sections have been obtained for elastic and inelastic scattering of protons by O16 (0>=Q>=-8.88 MeV). At laboratory angles of 80, 100, 120, 140, and 160°, continuous elastic excitation curves over the energy range from 13 to 19 MeV were obtained by a thick-target technique. For proton energies between 14.8 and 19.2 MeV, additional measurements with about 200-keV energy resolution were made by conventional thin-target techniques. Numerical values of cross sections for elastic and inelastic scattering are presented for the angular range from 15 to 160° at typically 10° angular and 200-keV energy intervals. Differential cross sections often change by a factor of 2 within 200-500 keV for large as well as small angles. The largest resonance is found at 14.9 MeV, but fluctuations at the upper end of the energy scale (~ 19 MeV), particularly for inelastic scattering, are not significantly smaller. Some, but not all of these strong fluctuations occur slightly above the five reaction thresholds (p, 2p), (p, d), (p, He3), (p, pn), (p, n) in the 12-to 19-MeV region. Previously, nine of the elastic angular distributions reported here had been made available for analysis by the use of an optical-model search code. Fits to within 10% were obtained, but the optical-model parameters for good fits showed strong energy dependence. It is suggested that simple potential scattering provides an insufficient explanation for O16(p, p)O16 for the energy region investigated. The interference of potential scattering with compound nuclear scattering is discussed and shown to be very noticeable, even for 200-keV energy resolution. Data and analysis emphasize the need for a theoretical approach to scattering which takes explicit account of easily formed states in light nuclei.