Physics-based numerical simulations of earthquakes and slow slip, coupled with field observations and laboratory experiments, can, in principle, be used to determine fault properties and potential fault behaviors. Because of the computational cost of simulating inertial wave-mediated effects, their representation is often simplified. The quasi-dynamic (QD) approach approximately accounts for inertial effects through a radiation damping term. We compare QD and fully dynamic (FD) simulations by exploring the long-term behavior of rate-and-state fault models with and without additional weakening during seismic slip. The models incorporate a velocity-strengthening (VS) patch in a velocity-weakening (VW) zone, to consider rupture interaction with a slip-inhibiting heterogeneity. Without additional weakening, the QD and FD approaches generate qualitatively similar slip patterns with quantitative differences, such as slower slip velocities and rupture speeds during earthquakes and more propensity for rupture arrest at the VS patch in the QD cases. Simulations with additional coseismic weakening produce qualitatively different patterns of earthquakes, with near-periodic pulse-like events in the FD simulations and much larger crack-like events accompanied by smaller events in the QD simulations. This is because the FD simulations with additional weakening allow earthquake rupture to propagate at a much lower level of prestress than the QD simulations. The resulting much larger ruptures in the QD simulations are more likely to propagate through the VS patch, unlike for the cases with no additional weakening. Overall, the QD approach should be used with caution, as the QD simulation results could drastically differ from the true response of the physical model considered.