We realize the growth of self-catalyzed core-shell GaAs/GaAsP nanowires (NWs) on Si substrates using molecular-beam epitaxy. Transmission electron microscopy of single GaAs/GaAsP NWs demonstrates their high crystal quality and shows domination of the GaAs zinc-blende phase. Using continuous-wave and time-resolved photoluminescence (PL), we make a detailed comparison with uncapped GaAs NWs to emphasize the effect of the GaAsP capping in suppressing the nonradiative surface states. Significant PL enhancement in the core-shell structures exceeding 3 orders of magnitude at 10 K is observed; in uncapped NWs PL is quenched at 60 K, whereas single core-shell GaAs/GaAsP structures exhibit bright emission even at room temperature. From analysis of the PL temperature dependence in both types of NW we are able to determine the main carrier escape mechanisms leading to the PL quench.