This work aims at comparing the damping performances of two passive damping treatments based on piezoelectric or viscoelastic patches. The motivation for such a comparison stems from the fact that the two damping techniques have been developed fairly independently, and are rarely compared. First, the dynamic response of a simply-supported metallic plate is measured experimentally after being equipped with constrained viscoelastic patches or piezoelectric patches connected to an electrical network. In order to extend the comparison, a numerical model of the structure is set up and validated to evaluate the damping performances of both passive treatments under different configurations (for instance equal-mass and equal-thickness configurations). Finally, with regard to these experimental and numerical results, the advantages and the limitations in using viscoelastic or piezoelectric treatments are discussed.