Colloidal semiconductor nanoplatelets with a similar electronic structure as quantum wells have recently emerged as exciting materials for optoelectronic applications. Here we investigate how morphology affects important photoluminescence properties of single CdSe and core/shell CdSe/CdZnS nanoplatelets. By analyzing photoluminescence intensity-lifetime correlation and second-order photon correlation results, we demonstrate that, irrespective of the morphology, Auger recombination plays only a minor role in dictating the blinking behavior of the nanoplatelets. We find that a rough shell induces additional nonradiative channels presumably related to defects or traps of an imperfect shell. Furthermore, polarization-resolved spectroscopy analysis reveals exciton fine-structure splitting of the order of several tens of meV in rough-shell nanoplatelets at room temperature, which is attributed to exciton localization and is substantiated by theoretical calculations taking into account the nanoplatelet shape and electron-hole exchange interaction.