The atomistic model improves substantially upon the continuum elasticity model in providing an accurate estimate for the Peierls stress. This is because it takes into account the change in the in-plane strain energy as the dislocation moves, as well as the misalignment energy.
However the model assumes that lattice resistance is dominating plastic flow and so cannot be used to predict τY. In materials (especially metals such as Al, Cu, Ni), the yield stress is substantially higher than τP. This is due to interactions with other obstacles and dislocation interactions, which this model does not consider. It is important to remember that we are considering only independent dislocations in this model; the calculations would become much more complicated if we consider dislocations interacting.
The model also does not take into account the anisotropy of crystals; the critical shear stress would depend on some combination of elastic constants, different for each plane. It would also be influenced by the anharmonic forces between atoms, which are here neglected, since the displacements near the dislocation line are large.
One application is in the toughening of non-metallic materials. Increasing toughness requires that the stress required for a dislocation to move must be substantially reduced. In this case we wish to decrease the energy associated with the misalignments across the slip plane, which act to reduce the dislocation width, w. These energies scale with the shear modulus.
The data from which the graphs were devised can be seen here.