Manifestations of non-adiabatic multiphonon transitions in optical properties of semiconductor quantum dots are theoretically analyzed accounting for the influence of the spherical confinement on the phonon energy spectrum. The role of phonon branches of different symmetry in the spectra of Raman scattering is ascertained. The effects of non-adiabaticity of the excit on-phonon system are shown to significantly enhance phonon-assisted transition probabilities and to lead to multiphonon optical spectra which are considerably different from the Franck-Condon progression. Calculated spectral progressions compare well with experimental data on photolumincscence and Raman scattering in CdSe and CdSe_xS_1-x quantum dots.