  Active Brownian particles (ABPs, such as self-phoretic colloids) swim at
fixed speed $v$ along a body-axis ${\bf u}$ that rotates by slow angular
diffusion. Run-and-tumble particles (RTPs, such as motile bacteria) swim with
constant $\u$ until a random tumble event suddenly decorrelates the
orientation. We show that when the motility parameters depend on density $\rho$
but not on ${\bf u}$, the coarse-grained fluctuating hydrodynamics of
interacting ABPs and RTPs can be mapped onto each other and are thus strictly
equivalent. In both cases, a steeply enough decreasing $v(\rho)$ causes phase
separation in dimensions $d=2,3$, even when no attractive forces act between
the particles. This points to a generic role for motility-induced phase
separation in active matter. However, we show that the ABP/RTP equivalence does
not automatically extend to the more general case of $\u$-dependent motilities.
