In this article we compare the two-photon photoluminescence and second harmonic generation from single connected gold nanodimers. Analyzing the particle size-dependent nonlinear optical spectra and performing excitation polarization resolved measurements using an experimental setup combining a femtosecond laser source with a parabolic mirror, we show that second harmonic generation and two-photon photoluminescence have different behaviors despite the same expected fundamental intensity-dependence. For further understanding of the observed phenomena, the plasmon resonances of single nanodimers are investigated using dark-field optical microscopy, and calculations are performed with Green's tensor method. Furthermore, the underlying mechanisms explaining the differences between these two optical processes are investigated using a surface integral equation method for the nonlinear computations. This study reveals that the different trends in the polarization-dependences of two-photon photoluminescence and second harmonic generation with the increasing diameters of the connected discs are due to their distinct physical nature, resulting in specific rules for plasmon enhancement and different coherence properties. Furthermore, this article clearly points out that special care has to be taken when two-photon photoluminescence and second harmonic generation are used to evaluate the amplitudes of electromagnetic hot-spots generated in plasmonic nanostructures.