We demonstrate that self-interacting dark matter models with interactions mediated by light particles can have significant deviations in the matter power-spectrum and detailed structure of galactic halos when compared to a standard cold dark matter scenario. While these deviations can take the form of suppression of small scale structure that are in some ways similar to that of warm dark matter, the self-interacting models have a much wider range of possible phenomenology. A long-range force in the dark matter can introduce multiple scales to the initial power spectrum, in the form of dark acoustic oscillations and an exponential cut-off in the power spectrum. Using simulations we show that the impact of these scales can remain observationally relevant up to the present day. Furthermore, the self-interaction can continue to modify the small-scale structure of the dark matter halos, reducing their central densities and creating a dark matter core. The resulting phenomenology is unique to this type of models.