The nuclear mean-field potentials obtained in the Hartree-Fock method with different choices of the in-medium nucleon-nucleon (NN) interaction have been used to study the equation of state (EOS) of the neutron star (NS) matter. The EOS of the uniform NS core has been calculated for the np$e\mu$ composition in the $\beta$-equilibrium at zero temperature, using version Sly4 of the Skyrme interaction as well as two density-dependent versions of the finite-range M3Y interaction (CDM3Y$n$ and M3Y-P$n$), and versions D1S and D1N of the Gogny interaction. Although the considered effective NN interactions were proven to be quite realistic in numerous nuclear structure and/or reaction studies, they give quite different behaviors of the symmetry energy of nuclear matter at supranuclear densities that lead to the \emph{soft} and \emph{stiff} scenarios discussed recently in the literature. Different EOS's of the NS core and the EOS of the NS crust given by the compressible liquid drop model have been used as input of the Tolman-Oppenheimer-Volkov equations to study how the nuclear symmetry energy affects the model prediction of different NS properties, like the cooling process as well as the gravitational mass, radius, and moment of inertia.