We estimate the diffuse supernova neutrino background (DSNB) using the recent progenitor-dependent, long-term supernova simulations from the Basel group and including neutrino oscillations at several post-bounce times. Assuming multi-angle matter suppression of collective effects during the accretion phase, we find that oscillation effects are dominated by the matter-driven MSW resonances, while neutrino-neutrino collective effects contribute at the 5-10% level. The impact of the neutrino mass hierarchy, of the time-dependent neutrino spectra and of the diverse progenitor star population is 10% or less, small compared to the uncertainty of at least 25% of the normalization of the supernova rate. Therefore, assuming that the sign of the neutrino mass hierarchy will be determined within the next decade, the future detection of the DSNB will deliver approximate information on the MSW-oscillated neutrino spectra. With a reliable model for neutrino emission, its detection will be a powerful instrument to provide complementary information on the star formation rate and for learning about stellar physics.