This paper studies the damping of slow surface MHD waves propagating along the equilibrium magnetic field on a finite-thickness magnetic interface. The plasma is assumed to be strongly magnetised, and the full Braginskii's expressions for viscosity and the heat flux are used. The primary focus of the paper is on the competition between resonant absorption in the thin dissipative layer embracing the ideal resonant position and the bulk wave damping due to viscosity and thermal conductivity as damping mechanisms for surface MHD waves. The dependence of the wave damping decrement on the wave length and the dissipative coefficients is studied. Application of the obtained results to the surface MHD wave damping in the solar chromosphere is discussed
