Quantum diffusion (QD) is studied via dissipative Madelung hydrodynamics. Initially, the wave packet spreads ballistically, then passes for an instant through normal diffusion and later tends asymptotically to a sub-diffusive law. It is shown that the apparent QD coefficient is not a universal physical parameter because it depends on the initial wave packet preparation. The overdamped QD of an electron in the field of a periodic potential is also investigated; in this case, the wave packet spreads logarithmically in time. Thermo-QD of heavier particles such as hydrogen, deuterium and tritium atoms in periodic potentials is studied and a simple estimate of the tunneling effect is obtained in the framework of a quasi-equilibrium semiclassical approach. The effective thermo-quantum temperature is also discussed in relation to the known temperature dependence of muon diffusivity in solids.