The influence of intrinsic chain stiffness on surface forces in solutions containing semiflexible polymers is investigated with density functional theory. The solvent is included explicitly, but comparisons are also made with exact results obtained with ideal chains, in the absence of solvent particles. We find that as the intrinsic stiffness increases, the polymers are at intermediate separations less able to form strong attractive bridges. This leads to a substantial free energy barrier. At short separations, bridging does dominate, resulting in an attractive interaction. However, the way in which the height of the free energy barrier responds to changes of the chain length, at a given stiffness, as well as on stiffness, for a given chain length, is nontrivial and not monotonic. There is furthermore a saturation effect, which at high monomer concentrations leads to a depletion interaction at intermediate separations. This amounts to a reversal of the mechanisms underlying the net interaction. That the density functional theory is accurate is verified by an excellent agreement between predicted structural properties and corresponding data obtained by simulations.