We present the results of a Monte Carlo study of the three-dimensional XY model and the three-dimensional antiferromagnetic three-state Potts model. In both cases we compute the difference of the free energies of a system with periodic and a system with antiperiodic boundary conditions in a neighborhood of the critical coupling. From the finite-size scaling behaviour of this quantity we extract values for the critical temperature and the critical exponent v that are compatible with recent high-statistics Monte Carlo studies of the models. The results for the free energy difference at the critical temperature and for the exponent v confirm that both models belong to the same universality class.