The functional dependence of excited-state geometries and normal modes calculated with time-dependent density functional theory (TDDFT) is investigated on the basis of vibronic structure calculations of the absorption spectra of large molecules. For a set of molecules covering a wide range of different structures including organic dyes, biological chromophores, and molecules of importance in material science, quantum mechanical simulations of the vibronic structure are performed. In total over 40 singlet-singlet transitions of neutral closed-shell compounds and doublet-doublet transitions of neutral radicals, radical cations, and anions are considered. Calculations with different standard density functionals show that the predicted vibronic structure critically depends on the fraction of the "exact" Hartree-Fock exchange (EEX) included in hybrid functionals. The effect can been traced back to a large influence of EEX on the geometrical displacement upon excitation. On the contrary, the dependence of the results on the choice of the local exchange-correlation functional is found to be rather small. On the basis of detailed comparisons with experimental spectra conclusions are drawn concerning the optimum amount of EEX mixing for a proper description of the excited-state properties. The relationship of the quality of the simulated spectra with the errors for 0-0 transition energies is discussed. For the investigated singlet-singlet pi-->pi* transitions and the first strongly dipole-allowed transitions of PAH radical cations some rules of thumb concerning the optimum portion of EEX are derived. However, in general no universal amount of EEX seems to exist that gives a uniformly good description for all systems and states. Nevertheless an inclusion of about 30-40% of EEX in the functional is found empirically to yield in most cases simulated spectra that compare very well with those from experiment and thus seems to be necessary for an accurate description of the excited-state geometry. Pure density functionals that are computationally more efficient provide less accurate spectra in most cases and their application is recommended solely for comparison purposes to obtain estimates for the reliability of the theoretical predictions.