Plasma carrying a spectrum of counterpropagating field-aligned ion-cyclotron waves can strongly and preferentially heat ions through a stochastic Fermi mechanism. Such a process has been proposed to explain the extreme temperatures, temperature anisotropies, and speeds of ions in the solar corona and solar wind. We quantify how differential flow between ion species results in a Doppler shift in the wave spectrum that can prevent this strong heating. Two critical values of differential flow are derived for strong heating of the core and tail of a given ion distribution function. Our comparison of these predictions to observations from the Wind spacecraft reveals excellent agreement. Solar wind helium that meets the condition for strong core heating is nearly 7 times hotter than hydrogen on average. Ion-cyclotron resonance contributes to heating in the solar wind, and there is a close link between heating, differential flow, and temperature anisotropy.