The mechanism of heating for hot, dilute, and turbulent plasmas represents a long-standing problem in space physics, whose implications concern both near-Earth environments and astrophysical systems. In order to explore the possible role of interparticle collisions, simulations of plasma turbulence—in both collisionless and weakly collisional regimes—have been compared by adopting Eulerian Hybrid Boltzmann–Maxwell simulations, being proton–proton collisions explicitly introduced through the nonlinear Dougherty operator. Although collisions do not significantly influence the statistical characteristics of the turbulence, they dissipate nonthermal features in the proton distribution function and suppress the enstrophy/entropy cascade in the velocity space, damping the spectral transfer toward large Hermite modes. This enstrophy dissipation is particularly effective in regions where the plasma distribution function is strongly distorted, suggesting that collisional effects are enhanced by fine velocity–space structures. A qualitative connection between the turbulent energy cascade in fluids and the enstrophy cascade in plasmas has been established, opening a new path to the understanding of astrophysical plasma turbulence.