Due to their central role in industrial formulations spanning from food packaging to smart coatings, polymer nanocomposites have been the object of a remarkable attention over the last two decades. Incorporating nanoparticles (NPs) to a polymer matrix modifies the conformation and mobility of the polymer chains at the NP-polymer interface and can potentially provide materials with enhanced properties as compared to pristine polymers. To this end, it is crucial to predict and control the ability of NPs to diffuse and achieve a good dispersion in the polymer matrix. Understanding how to control the NPs' dispersion is a challenging task controlled by the delicate balance between enthalpic and entropic contributions, such as NP-polymer interaction, NP size and shape, and polymer chain conformation. By performing Molecular Dynamics (MD) simulations, we investigate the effect of polymer chains' stiffness on the mobility of spherical NPs that establish weak or strong interactions with the polymer. Our results show a sound dependence of the NPs' diffusivity on the long-range order of the polymer melt, which undergoes an isotropic-to-nematic phase transition upon increasing chain stiffness. This phase transition induces a dynamical anisotropy in the nematic phase, with the NPs preferentially diffusing along the nematic director rather than in the directions perpendicular to it. Not only does this tendency determine the NPs' mobility and degree of dispersion in the polymer matrix, but it also influences the resistance to flow of the polymer nanocomposite when a shear is applied. In particular, to assess the role of the chains' conformation on the macroscopic response of our model PNC, we employ reverse non-equilibrium MD to calculate the zero-shear viscosity in both the isotropic and nematic phases, and unveil a plasticizing effect at increasing chain stiffness when the shear is applied along the nematic axis.