The numerical stability, equilibrium diffusive limit, and accuracy of the variable Eddington factor (VEF) methods and flux-limited diffusion methods for radiation transport calculations are considered. The diffusive limit analysis proves that three VEF closures and their associated flux-limiters retain full first-order accuracy in the equilibrium diffusion limit while achieving the correct propagation speed in the optically thin streaming limit. The stability analysis reveals that the flux-limited diffusion methods are unconditionally stable, but the VEF equations with an arbitrary nonlinear closure can be numerically unstable for certain commonly used differencing schemes. However, regular solutions to the VEF equations are obtainable by Godunov-type schemes. Numerical comparisons among various solutions for a test problem show that flux-limited diffusion methods are only slightly less accurate than their corresponding VEF methods, and the Minerbo VEF method and the Minerbo flux-limited diffusion method are in general more accurate than other approximations.