The effects of contact geometry and specific contact resistivity on In0.53Ga0.47As (InGaAs) and silicon (Si) nanoscale (18 nm channel length) n-channel FinFETs performance, and the effects of models thereof, are studied using a quantum-corrected semiclassical Monte Carlo method. Saddle/slot, raised source and drain (RSD), and reference end contacts are modeled. Both ideal perfectly injecting and absorbing contacts and those with more realistic specific contact resistivities are considered. Far-from-equilibrium degenerate statistics, quantum-confinement effects on carrier distributions in real-space and among energy valleys and on scattering, and quasiballistic transport are modeled. Silicon ⟨110⟩ channel and Si ⟨100⟩ channel FinFETs, multivalley InGaAs channel FinFETs with conventionally reported InGaAs energy valley offsets, and reference idealized Γ-valley-only InGaAs (Γ-InGaAs) channel FinFETs are simulated. Among our findings, InGaAs channel FinFETs are highly sensitive to modeled contact geometry and specific contact resistivity and to the band structure model, while Si channel FinFETs showed still significant but much less sensitivity to the contact models. For example, for idealized unity transmissivity contacts, Γ-InGaAs channel FinFETs performed best for all contact geometries, at least in terms of transconductance, and end contacts provided the best performance for all considered channel materials. For realistic contact resistivities, however, the results are essentially reversed. Silicon channel FinFETs performed best for all contact geometries, and saddle/slot and RSD contacts outperformed end contacts.