We investigate optical absorption spectra obtained through time-dependent density functional theory (TD-DFT) based on nonempirical hybrid functionals that are designed to correctly reproduce the dielectric function. The comparison with state-of-the-art GW calculations followed by the solution of the Bethe-Salpeter equation (BSE-GW) shows close agreement for both the transition energies and the main features of the spectra. We confront TD-DFT with BSE-GW by focusing on the model dielectric function and the local exchange-correlation kernel. The present TD-DFT approach achieves the accuracy of BSE-GW at a fraction of the computational cost.