Solving the mass gap problem for a field theory, means proving that there is an energy gap between the ground state and the first excited state. (The alternative situation is where there is a continuum of excited states with energies approaching arbitrarily close to zero.) For example, if you could prove that glueballs are the lightest bound state in QCD, and have a mass greater than zero.
In the electroweak theory, the Higgs field is something extra that you introduce, in addition to the gauge fields, in order to indirectly give the gauge bosons mass. (The alternative here would be for the gauge fields to be defined as massive from the beginning, by having a mass term in their Lagrangian.)
So it's the difference between proving that a theory already has mass gap, and modifying it so that it has a mass gap as a result of the modification. You actually could add a Higgs field with color charge to QCD, and thereby give the gluons mass, but that's not the problem.