In this work, aspects of discretization errors associated with finite volume (FV) and equivalent finite element (FE) modelling strategies are discussed within the framework of polymer melt flow. The computational approaches are based on the generalized Newtonian model in conjunction with Cross constitutive equation. The numerical examples illustrate one and two-dimensional fluid flows, in which the latter is discretized using structured quadrilateral elements / volumes. A study on the best strategy to compute non-linear viscosities at control volume boundaries is also presented for FV. Based on well established a posteriori error estimation techniques, it is demonstrated that, in this class of problems, FV discretization errors and differences between FE and FV solutions are greatly affected by the scheme used to compute the FV non-linear coefficients at the control volume surfaces. Simulations for rectangular channels show that FE yields smaller global errors then FV for velocity and temperature solutions.