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