Two-photon excitation in fluorescence correlation spectroscopy (FCS) is often preferred to one-photon excitation because of reduced bulk photobleaching and photodamage, and deeper penetration into scattering media, such as thick biological specimens. Two-photon FCS, however, suffers from lower signal-to-noise ratios which are directly related to the lower molecular brightness achieved. We compare standard FCS with a fixed measurement volume with scanning FCS, where the measurement volume is scanned along a circular path. The experimental results show that photobleaching is the dominant cause of the effects observed at the high excitation powers necessary for good signal-to-noise ratios. Theoretical calculations assuming a non-uniform excitation intensity profile, and using the concept of generalized volume contrast, provide an explanation for the photo-bleaching effects commonly observed in two-photon FCS at high excitation intensities, without having to assume optical saturation. Scanning alleviates these effects by spreading the photobleaching dose over a larger area, thereby reducing the depletion of fluorescent molecules in the measurement volume. These results, which facilitate understanding of the photobleaching in FCS and of the positive effects of scanning, are particularly important in studies involving the autocorrelation amplitude g(0), such as concentration measurements or binding studies using fluorescence cross-correlation between two labeled species.