There is a general consensus among archaeologists that replacement of Neanderthals by anatomically modern humans in Europe occurred around 40K to 35K YBP. However, the causal mechanism for this replacement continues to be debated. Searching for specific fitness advantages in the archaeological record has proven difficult, as these may be obscured, absent, or subject to interpretation. Proposed models have therefore featured either fitness advantages in favor of anatomically modern humans, or invoked neutral drift under various preconditions. To bridge this gap, we rigorously compare the system-level properties of fitness- and drift-based explanations of Neanderthal replacement. Our stochastic simulations and analytical predictions show that, although both fitness and drift can produce replacement, they present important differences in 1) required initial conditions, 2) reliability, 3) time to replacement, and 4) path to replacement (population histories). These results present useful opportunities for comparison with archaeological and genetic data. We find far greater agreement between the available empirical evidence and the system-level properties of replacement by differential fitness, rather than by neutral drift.