Off-lattice Monte Carlo computer simulations and numerical polymer reference interaction site model (PRISM) integral equation calculations were performed to quantitatively probe the origins of entropic corrections to Flory-Huggins theory for athermal polymer blends with stiffness disparity. This model system is of interest since it has been recently proposed for describing commercially relevant hydrocarbon polymer mixtures. The novelty of the simulations is that the chemical potential changes on mixing for both components are evaluated. We have considered mixing under constant density conditions, and find surprisingly that the stiffer component is stabilized on blending, while the flexible component is characterized by a positive interaction or chi parameter. The net effective single chi parameter describing these blends, however, is close to zero suggesting that they are completely miscible over a wide range of stiffness disparities and chain lengths. PRISM theory is found to be in good agreement with the simulations for both structural and mixing thermodynamic properties. While purely entropic nonrandom mixing effects could be relevant in determining system thermodynamics, especially for large stiffness disparity, the dominant contribution to the chemical potential changes on mixing arise from equation-of-state (EOS) effects since the two pure components and the mixture are at different pressures when examined at the same density. The EOS contribution to the mixing free energy for small stiffness mismatch is shown to be quantitatively reproduced through an extension of the generalized Flory approach. Through the use of PRISM theory we find that athermal, nonlocal entropy-driven phase separation can occur for long enough chains and high enough stiffness disparity. However, since no phase separation is predicted for stiffness disparities relevant to experimental hydrocarbon systems, regardless of chain length, we suggest that enthalpic effects have to be evoked to explain the limited miscibility of these commercially important mixtures.