Dependence of optical properties of gold solid nanospheroids and spheroidal nanoshells on the dielectric environment has been studied in comparison with standard spherical models, i.e. with solid spheres and spherical nanoshells, respectively. The extinction and light scattering spectra were calculated using Mie theory, T-matrix, and the separation of variables methods. The sensitivity of plasmon resonance (PR) tuning to variations of the refractive index (n=1.3-l.7) was studied for particles with different equivolume size. In the case of nanoshells, the metal layer thickness was also varied. For nanoparticles with 15-nm equivolume diameter, the maximal PR shifts are observed in the case of spheroidal nanoshells and the calculated PR shifts decrease in the order: spheroidal nanoshells, solid spheroids, and spherical nanoshells. For particles with the 60-nm equivolume diameter, almost equal PR shifts are observed both for solid spheroids and spherical nanoshells, whereas the spheroidal nanoshells exhibit the maximal PR shifts. Other things being equal, the PR of silver nanoparticles is more sensitive to the dielectric environment as compared to the gold counterparts.