The viscosity solutions was first introduced in the context of the Hamilton-Jacobi equation by the vanishing viscosity method. It would be difficult to apply the notion of distributional weak solution in this context, because the derivatives occur inside a nonlinear function. Further complications would arise because a distributional weak solution is not necessarily unique (for degenerate elliptic equations or nonlinear first order equations). Because of the nonlinearity, a Hamilton-Jacobi equations often doesn't have a classical solution, even if the Hamiltonian is an analytic function.
The viscosity solution was later generalized to degenerate elliptic equations, where the vanishing viscosity method itself no longer works. In case the equation is linear in the derivatives, both distributional weak solutions and viscosity solutions are defined, and we can investigate their relationship. My guess would be that the viscosity solutions turns out to also be a weak solutions in this context, but that there will in general be also weak solutions which are not viscosity solutions.