This study is concerned with the modelling of thixotropic and viscoelastoplastic material systems, contrasting two approaches in the development of such constitutive models. Accordingly, departure from Oldroyd-Blike behaviour is engineered through, first, a new micellar viscoelastic-thixotropic model (NM_tau(p)_ABS), under the Bautista-Manero framework, and second, a De Souza model. This NM_tau(p)_ABS model, is based on the energy dissipated by a micellar material to change its internal structure, whilst equivalently, the De Souza model employs the second invariant of stress. These models are compared and contrasted in their response through counterpart numerical solutions for axisymmetric contraction-expansion flow. Here, solution features of yielded-unyielded regions, total pressure drop, stress fields and vortex dynamics are analysed under scaling based on the second-Newtonian viscosity-plateau (eta(s)). With the NM_tau(p)_ABS model, yield-stress features are identified through solvent-fraction beta-variation. In contrast, for the De Souza model, counterpart yield-stress features are exposed through yield-stress t(0d)-variation. With either yield-stress increase or rise in elasticity, NM_tau(p)_ABS solution response appears symmetrical about the contraction-plane axis, whilst De Souza patterns prove asymmetrical. Under solvent-fraction decrease, NM_tau(p)_ABS response provides yielded-region shrinkage, upstream and downstream vortex suppression, and non-zero N-1-region growth. Moreover, under elasticity rise, fading non-zero N-1-regions, size-invariant yield-fronts and non-zero N-1-regions are observed. In contrast under tau(0d) increase or rise in elasticity, De Souza solutions manifest enhancement in vortex activity, and non-zero N-1-region-intensification and expansion. Furthermore, tau(0d)-rise provokes De Souza yielded-region shrinkage, whilst elasticity does the opposite. On total pressure drop (Delta p), for NM_tau(p)_ABS and with polymeric-fraction increase at fixed Wi, both monotonic rise at low-Wi and decline at higher-Wi are gathered. In contrast, only a monotonic rising trend is recorded with De Souza Delta p-solutions for fixed Wi under t(0d)-rise. Furthermore, with Wi-rise and at any fixed t(0eff)-level, both models concur in a declining Delta p-trend.