The resolution and the robustness of the weighted essentially non-oscillatory (WENO) scheme and two-step finite-difference WENO (TSFDWENO) schemes are compared by strictly using the same flux evaluation method and smoothness indicators. TSFDWENO schemes are defined to include a family of weighted compact nonlinear scheme (WCNS) and an alternative WENO scheme. Comparison results indicate that WCNS has a higher resolution than the WENO scheme, while the WENO scheme is more robust than WCNS. Additionally, various flux evaluation methods are combined with TSFDWENO schemes, and they are evaluated. Then, the effects of the flux evaluation methods on the resolution and robustness of the scheme are investigated, and the results show that the robustness and the resolution can be significantly altered by changing the flux evaluation method. This study reveals the advantage of being able to use various flux evaluation methods in the TSFDWENO scheme as well as the fair comparison of the WENO schemes and WCNS. On the other hand, these effects are marginalized when changing the interpolation and differencing method. Such knowledge can be important when selecting schemes for actual simulation and developing guidelines for scheme improvement.