We study the diversities in the properties of the neutron stars arising due to the different choices for the cross coupling between various mesons, which governs the density dependence of the nuclear symmetry energy in the extended relativistic mean-field (RMF) model. For this purpose, we obtain two different families of the extended RMF model corresponding to different nonlinear cross-coupling terms in the isovector part of the effective Lagrangian density. The lowest-order contributions for the δ mesons are also included. The different models within the same family yield wide variation in the value of neutron-skin thickness in the Pb208 nucleus. These models are employed to compute the neutron-star properties such as core-crust transition density, radius and red shift at canonical mass (1.4M), tidal polarizability parameter, and threshold mass required for the enhanced cooling through the direct Urca process. Most of the neutron-star properties considered are significantly different(10-40%) for the different families of models at a smaller neutron-skin thickness (∼0.15fm) in the Pb208 nucleus.