A thorough understanding of properties of neutron stars requires both a reliable knowledge of the equation of state (EOS) of super-dense nuclear matter and the strong-field gravity theories simultaneously. To provide information that may help break this EOS-gravity degeneracy, we investigate effects of nuclear symmetry energy on the gravitational binding energy of neutron stars within GR and the scalar-tensor subset of alternative gravity models. We focus on effects of the slope L of nuclear symmetry energy at saturation density and the high-density behavior of nuclear symmetry energy. We find that the variation of either the density slope L or the high-density behavior of nuclear symmetry energy leads to large changes in the binding energy of neutron stars. The difference in predictions using the GR and the scalar-tensor theory appears only for massive neutron stars, and even then is significantly smaller than the difference resulting from variations in the symmetry energy.