Catanionic vesicles are formed spontaneously by mixing cationic and anionic dispersions in aqueous solution in suitable conditions. Because of spontaneity in formation, long-term stability, and easy modulation of size and charge, they have numerous advantages over conventional lipid based vesicles. The dynamics of such vesicles is of interest in the field of biomedicine, as they can be used to deliver drug molecules into the cell membrane. Dynamics of catanionic vesicles based on sodium dodecyl sulfate (SDS) and cetyltrimethylammonium bromide (CTAB) have been studied using incoherent elastic and quasielastic neutron scattering (QENS) techniques. Neutron scattering experiments have been carried out on two backscattering spectrometers, IRIS and IN16B, which have different energy resolutions and energy transfer windows. An elastic fixed-window scan carried out using IN16B shows a phase transition at similar to 307 K during the heating cycle, whereas on cooling the transition occurred at similar to 294 K. DSC results are found to be in close agreement with the elastic scan data. This transition is ascribed to a structural rearrangement from a multilamellar to a unilamellar phase [Andreozzi et al. J. Phys. Chem. B 2010, 114, 8056-8060]. It is found that a model in which the surfactant molecules undergo both lateral and internal motions can describe the QENS data quite well. While the data from IRIS have contributions from both dynamical processes, the data from IN16B probe only lateral motions, as, the internal motions are too fast for the energy window of the spectrometer. It is found that, through the transition, the fraction of surfactant molecules undergoing lateral motion increases of a factor of 2 from the multilamellar to the unilamellar phase, indicating an enhanced fluidity of the latter. The lateral motion is found to be Fickian in nature, while the internal motion has been described by a localized translational diffusion model. The results reported here could have direct interest for a number of applications, such as molecular transport, and the effect of specific drug molecules or hormones through the membrane.