In the present paper we consider two natural, elliptic, self-adjoint second order differential operators acting on exterior differential forms on Riemannian manifolds. These operators are the wellknown Hodge-de Rham and little-known Tachibana operators. Basic properties of these operators are very similar, or vice versa are dual with respect to each other. We review the results (partly obtained by the authors) on the geometry of these operators and demonstrate the comparative analysis of their properties.