Themain object of investigation inmagnonics, spin waves (SWs) are promising information carriers. Presently, the most commonly studied are plane-wave-like SWs and SWs propagating in confined structures, such as waveguides. Here we consider a Gaussian SW beam obliquely incident on an ultranarrow interface between two identical ferromagnetic materials. We use an analytical model and micromagnetic simulations for an in-depth analysis of the influence of the interface properties, in particular the magnetic anisotropy, on the transmission of the SW beam. We derive analytical formulas for the reflectance, transmittance, phase shift, and Goos-Hanchen (GH) shift for beams reflected and refracted by an interface between two semi-infinite ferromagnetic media. The GH shifts in SW beam reflection and transmission are confirmed by micromagnetic simulations in the thin-film geometry. We demonstrate the dependence of the characteristic properties on the magnetic anisotropy at the interface, the angle of incidence, and the frequency of the SWs. We also propose a method for the excitation of high-quality SW beams in micromagnetic simulations.