The random walk web distance is a natural directed distance on the trajectory of coalescing simple random walks. It is given by the number of jumps between different random walk paths when one is only allowed to move in one direction. The Brownian web distance is the scale-invariant limit of the random walk web distance. It is integer-valued and has scaling exponents 0:1:2 as compared to 1:2:3 in the KPZ world.