To properly describe heating in weakly collisional turbulent plasmas such as the solar wind, interparticle collisions should be taken into account. Collisions can convert ordered energy into heat by means of irreversible relaxation towards the thermal equilibrium. Recently, Pezzi et al. (Phys. Rev. Lett., vol. 116, 2016a, 145001) showed that the plasma collisionality is enhanced by the presence of fine structures in velocity space. Here, the analysis is extended by directly comparing the effects of the fully nonlinear Landau operator and a linearized Landau operator. By focusing on the relaxation towards the equilibrium of an out of equilibrium distribution function in a homogeneous force-free plasma, here it is pointed out that it is significant to retain nonlinearities in the collisional operator to quantify the importance of collisional effects. Although the presence of several characteristic times associated with the dissipation of different phase space structures is recovered in both the cases of the nonlinear and the linearized operators, the influence of these times is different in the two cases. In the linearized operator case, the recovered characteristic times are systematically larger than in the fully nonlinear operator case, this suggesting that fine velocity structures are dissipated more slowly if nonlinearities are neglected in the collisional operator.