Interfacial instabilities of superposed pressure driven channel flow of viscoelastic fluids are investigated theoretically using linear stability analysis. Nonlinear constitutive equations which accurately depict the steady as well as transient viscoelastic properties of typical polymeric melts and solutions with various degrees of flexibility and accuracy are used to assess the constitutive complexity required to accurately describe the stability characteristics of this class of flows by comparing the results of the stability analysis with the experimental results of Wilson and Khomami (J. Non. Newtonian Fluid Mech. 41 (1992) 255; J. Rheol. 37 (1993) 315) and Khomami and Ranjbaran (Rheol. Acta 36 (1997) 1-22). It is shown that the multimode Giesekus model, which can accurately describe the steady as well transient behavior of the polymeric test fluids used in the experiments, can quantitatively describe the interfacial instability phenomenon in terms of the neutral stability contour as well as the growth/decay rate behavior. A rigorous energy analysis based on a newly developed disturbance-energy equation for viscoelastic flows is performed to investigate the mechanism of the purely viscous and purely elastic interfacial instabilities in pressure and drag driven channel flows. The mechanism of shortwave purely viscous instability is found to be due to the viscosity mismatch and the subsequent perturbation vorticity mismatch at the interface (i.e. interfacial friction), whereas the mechanism of the longwave purely viscous instability is found to be due to the bulk Reynolds stresses. The mechanism of purely elastic instability is found to be due to the coupling between the perturbation velocity and the jump in normal stresses across the interface at longwaves as well as shortwaves. An examination of perturbation velocity field reveals that for purely elastic longwave instability the jump in the normal stresses across the interface leads to a perturbation back flow in the bulk resulting in either accumulation (destabilizing) or depletion (stabilizing) of the fluid below the crest of the perturbed interface. In the case of purely elastic shortwave instability, coupling of the jump in the normal stresses across the interface and the perturbation velocity leads to perturbation vorticities adjacent to the interface which drive the instability.