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We study the traffic of two types of molecular motors using the two-species asymmetric simple exclusion process (ASEP) with periodic boundary conditions and with attachment and detachment of particles. We determine characteristic properties such as motor densities and currents by simulations and analytical calculations. For motors with different unbinding probabilities, mean-field theory gives the correct bound density and total current of the motors, as shown by numerical simulations. For motors differing in their stepping probabilities, the particle-hole symmetry of the current-density relationship is broken and mean-field theory fails drastically. The total motor current exhibits exponential finite-size scaling, which we use to extrapolate the total current to the thermodynamic limit. Finally, we also study the motion of a single motor in the background of many nonmoving motors.