Both remote sensing and in situ measurements show that the fast solar wind plasma significantly deviates from thermal equilibrium and is strongly permeated by turbulent electromagnetic waves, which regulate the ion temperature anisotropies and relative drifts. Thus, the ion kinetics is governed by heating and cooling related to absorption and emission of ion‐acoustic and ion‐cyclotron waves, as well as nonresonant pitch angle scattering and diffusion in phase space. Additionally, the solar wind properties are affected by its nonadiabatic expansion as the wind travels away from the Sun. In this study we present results from 1.5‐D hybrid simulations to investigate the effects of a nonlinear turbulent spectrum of Alfvén‐cyclotron waves and the solar wind expansion on the anisotropic heating and differential acceleration of protons and He++ ions. We compare the different heating and acceleration by turbulent Alfvén‐cyclotron wave spectra and by pure monochromatic waves. For the waves and the wave spectra used in our model, we find that the He++ ions are preferentially heated and by the end of the simulations acquire much more than mass‐proportional temperature ratios, Tα/Tp>mα/mp. The differential acceleration between the two species strongly depends on the initial wave amplitude and the related spectral index and is often suppressed by the solar wind expansion. We also find that the expansion leads to perpendicular cooling for both species, and depending on the initial wave spectra, it can either heat or cool the ions in parallel direction. Despite the cooling effect of the expansion in perpendicular direction, the wave‐particle interactions provide an additional heating source, and the perpendicular temperature components remain higher than the adiabatic predictions. Key Points He++ ions heating and acceleration by Alfven-cyclotron wave-spectra Solar wind expansion ‐ perpendicular cooling, parallel heating with wave-spectra Ion trapping results in prominent proton and He++ ion beam formations