Semiconductor nanocrystals consisting of a quantum dot (QD) core and a quantum well (QW) shell, where the QD and QW are separated by a tunneling barrier, offer a unique opportunity to engineer the photophysical properties of individual nanostructures. Using the thicknesses of the corresponding layers, the excitons of the first and second excited states can be separated spatially, localizing one state to the QD and the other to the QW. Thus the wave function overlap of the two states can be minimized, suppressing non-radiative thermalization between the two wells, which in turn leads to radiative relaxation from both states. The molecular analogy to such dual emission would be the inhibition of internal conversion, a special case that violates Kasha's rule. Using nanosecond time-resolved spectroscopy of QDQW CdSe/ZnS onion-like nanocrystals, an intermediate regime of exciton separation and suppressed thermalization is identified where the non-radiative relaxation of the higher-energy state is slowed, but not completely inhibited. In this intermediate thermalization regime, the temporal evolution of the delayed emission spectra resulting from trapped carriers mimic the dynamics of such states in nanocrystals that consist of only a QD core. In stark contrast, when a higher-energy metastable state exists in the QW shell due to strongly suppressed interwell thermalization, the spectral dynamics of the long-lived excitations in the QD and QW, which are spectrally distinct, are amplified and differ from each other as well as from those in the core-only nanocrystals. This difference in spectral dynamics demonstrates the utility of exploiting well-defined exciton localization to study the nature and spatial dependence of the intriguing photophysics of colloidal semiconductor nanocrystals, and illustrates the power of nanosecond gated luminescence spectroscopy in illuminating complex relaxation dynamics which are entirely masked in steady-state or ultrafast spectroscopy.