This study focuses on the parametrical investigation and comparison of different non-Newtonian effects due to polymer/surfactant additives on the flow structure around a circular cylinder in periodic vortex shedding regime. For this purpose, inelastic power-law and viscoelastic Oldroyd-B, Giesekus and FENE-P models are adopted in the constitutive formulation. The range of the Reynolds number is 80 <= Re <= 300 the power-law index range is 0.6 <= n <= 1.2 and the Weissenberg number range is 0 <= Wi <= 1.2. A finite-element based solver is used in the two dimensional flow simulations. The resulting flow structure is compared and discussed in terms of the variation in drag and lift coefficients, vortex shedding frequency, vortex formation length, separation angle and the critical Reynolds number for the onset of vortex shedding. Drag reduction is observed for shear thinning fluids as also observed in previous studies with inelastic models; moreover, the vortex shedding frequency increases, and both the vortex formation length and separation angle decrease under shear thinning. Drag reduction in shear thinning fluids is mainly due to reduction in friction drag. Fluid elasticity leads to an increase in the drag coefficient and vortex formation length, and a decrease in the vortex shedding frequency and separation angle. Also, anisotropy effects through the mobility factor decrease the drag, the vortex formation length, and increases the shedding frequency. The extensibility of polymer molecules increases the drag, the vortex formation length, and decreases the shedding frequency. It is also observed that weakly elastic effects have almost no impact on the onset of periodic vortex shedding.