  The Hall coefficient $R_H$ of Sr$_2$RuO$_4$ exhibits a non-monotonic
temperature dependence with two sign reversals. We show that this puzzling
behavior is the signature of two crossovers which are key to the physics of
this material. The increase of $R_H$ and the first sign change upon cooling are
associated with a crossover into a regime of coherent quasiparticles with
strong orbital differentiation of the inelastic scattering rates. The eventual
decrease and the second sign change at lower temperature is driven by the
crossover from inelastic to impurity-dominated scattering. This qualitative
picture is supported by quantitative calculations of $R_H(T)$ using Boltzmann
transport theory in combination with dynamical mean-field theory, taking into
account the effect of spin-orbit coupling. Our insights shed new light on the
temperature dependence of the Hall coefficient in materials with strong orbital
differentiation, as observed in Hund's metals.
