  In this paper, we give a new proof for the fact that the distributional weak
solutions and the viscosity solutions of the $p$-Laplace equation
$-\diver(\abs{Du}^{p-2}Du)=0$ coincide. Our proof is more direct and
transparent than the original one by Juutinen, Lindqvist and Manfredi
\cite{jlm}, which relied on the full uniqueness machinery of the theory of
viscosity solutions. We establish a similar result also for the solutions of
the non-homogeneous version of the $p$-Laplace equation.
