Based on principles of nonequilibrium thermodynamics, we derive generalized differential constitutive equations for polymer melts which incorporate terms that account for anisotropic hydrodynamic drag in the form suggested by Giesekus, finite chain extensibility with non-linear molecular stretching, non-affine deformation, and variation of the longest chain relaxation time with chain conformation. In the new equations, the expression for the Helmholtz free energy of deformation is defined such that the entropy remains bounded even at high deformation rates, as it should from a physical point of view. Key elements in the new constitutive models are the functions describing the dependence of the nonequilibrum free energy and relaxation matrix on the conformation tensor. Restrictions on the parameters entering these two functions are obtained by analyzing the thermodynamic admissibility of the model. With suitable choices of these two functions, the new equations reduce to a number of well-known viscoelastic models. However, they are more general in the sense that they permit incorporating into a single constitutive differential equation more accurate expressions for the description of chain elasticity and relaxation. The new equations are used to describe (fit) rheological data provided either by experimental measurements on industrial samples or obtained through Non-Equilibrium Molecular Dynamics (NEMD) simulations in shear and planar elongation.