In the context of the decisive role that vibronic interactions play in the functioning of molecular quantum cellular automata, in this article we give a comparative analysis of the two alternative vibronic approaches to the evaluation of the key functional characteristics of molecular cells. Semiclassical Born-Oppenheimer approximation and quantum mechanical evaluations of the vibronic energy pattern, electronic density distributions and cell-cell response function are performed for two-electron square-planar mixed valence molecular cells subjected to the action of a molecular driver. Special emphasis is put on the description of the cell-cell response function, which describes strong non-linearity as a prerequisite for the effective action of quantum cellular automata. Comparison of results obtained within the semiclassical and quantum-mechanical approaches has revealed a drastic difference between the shapes of the cell-cell response functions evaluated within these two approaches in the case of moderate vibronic coupling when the energy levels of the square cell interacting with a weakly polarized driver undergo large tunneling splitting in shallow adiabatic potential minima. In contrast, in the limits of strong vibronic coupling (a double-well adiabatic potential with deep minima) and weak vibronic coupling (a single well adiabatic potential) the adiabatic approximation is shown to describe the cell-cell response function with rather good accuracy.