The spatiotemporal clustering of earthquakes is a feature of medium- and short-term seismicity, indicating that earthquakes interact. However, controversy exists about the physical mechanism behind aftershock triggering: static stress transfer and reloading by postseismic processes have been proposed as explanations. In this work, we use a Coulomb rate-and-state model to study the role of coseismic and postseismic stress changes on aftershocks and focus on two processes: creep on the main shock fault plane (afterslip) and secondary aftershock triggering by previous aftershocks. We model the seismic response to Coulomb stress changes using the Dieterich constitutive law and focus on two events: the Parkfield, Mw 6.0, and the Tohoku, Mw 9.0, earthquakes. We find that modeling secondary triggering systematically improves the maximum log likelihood fit of the sequences. The effect of afterslip is more subtle and difficult to assess for near-fault events, where model errors are largest. More robust conclusions can be drawn for off-fault aftershocks: following the Tohoku earthquake, afterslip promotes shallow crustal seismicity in the Fukushima region. Simple geometrical considerations indicate that afterslip-induced stress changes may have been significant on trench parallel crustal fault systems following several of the largest recorded subduction earthquakes. Moreover, the time dependence of afterslip strongly enhances its triggering potential: seismicity triggered by an instantaneous stress change decays more quickly than seismicity triggered by gradual loading, and as a result we find afterslip to be particularly important between few weeks and few months after the main shock.