Unequal twins: Probability distributions do not determine everything

It is the common lore to assume that knowing the equation for the probability distribution function (PDF) of a stochastic model as a function of time tells the whole picture defining all other characteristics of the model. We show that this is not the case by comparing two exactly solvable models of anomalous diffusion due to geometric constraints: the comb model and the random walk on a random walk. We show that though the two models have exactly the same PDFs, they differ in other respects, like their first passage time distributions, their autocorrelation functions, and their aging properties.