Gamma-ray burst (GRB) X-ray light curves exhibit a variety of complex temporal structures, such as flares and plateaus. The origin of flares seen in many GRB early afterglows is still uncertain. Here we analyze a sample of 89 GRBs, 61 of them with flares, both with and without a “plateau” phase. We fit the Swift X-Ray Telescope light curves with synchrotron emission from a forward shock propagating into either a constant-density interstellar medium or a stellar wind and flares on top of that. We find that the flare light curves are not symmetric, with a decay time that is approximately three to four times longer than the rise time. We do not find any differences in flare properties between GRBs with and without a “plateau” phase. Moreover, additional afterglow properties such as the electron power-law index and the end time of the plateau are consistent between bursts with and without flares. These results strongly indicate that flares originate from a mechanism distinct from that producing the plateau and afterglow. When looking at the prompt-emission properties, we do find some tendencies: GRBs with flares tend to be brighter and longer lasting than GRBs without flares. We therefore conclude that, unlike plateaus, flares are unlikely to arise from an external origin and are more plausibly associated with prolonged central engine activity that lasts longer than the main episode that produces the prompt phase. While the plateau cannot share the same origin, our results indicate that it is unlikely to be dominated by late-time accretion episodes associated with flaring activity, although alternative forms of late-time energy injection remain possible.