We present a comparative analysis of tight-binding and free-electron calculations of the conductance of an atomic-scale metallic contact. The calculations are based on a full dynamic simulation of the atomic structure during the pulloff of the contact, for a range of temperatures. As in previous simulations, we find that the contact evolves through a series of mechanical instabilities and can become highly disordered prior to fracture. Both the mechanical evolution of the contact and the behavior of the conductance depend strongly on temperature. We find that conductance quantization is destroyed easily by irregularities in the shape of the contact and, in the tight-binding model, also in the internal atomic structure of the contact. In the tight-binding calculation conductance quantization is seen only at high temperature, when the contact geometry and structure become very regular. With the free-electron model, we see perfectly quantized conductance plateaus just prior to contact fracture, while the plateaus in the earlier history of the contact are washed out by tunneling. In the free-electron calculation, conductance quantization is seen both at low and at high temperature but is more prominent at high temperature. We use the tight-binding and free-electron results for the conductance to obtain a calibration curve relating the conductance to the constriction width. The calculated conductances lie significantly below the Sharvin limit but the inclusion of the first-order semiclassical correction to the Sharvin formula greatly improves the agreement.