Ultrathin Fe films on Pt substrates have been investigated under ultrahigh vacuum conditions by scanning tunneling microscopy, low energy electron diffraction, magneto-optical Kerr effect, x-ray magnetic circular dichroism measurements, and Kerr microscopy. We present a comparison between Fe films on flat Pt(111) and stepped Pt(997), with particular focus on the magnetic anisotropy in the submonolayer thickness range below 0.2 monolayer coverage, and above the spin reorientation transition at 3 monolayer thickness. The comparison of structure and magnetism suggests that the perpendicular easy axis found for films thinner than three monolayers is due to dominating contributions from both film interfaces to the anisotropy energy. The Fe-Pt interface contribution has its origin in the hybridization of the Fe 3d with the Pt 5d band. The in-plane magnetic anisotropy above 3 atomic layers film thickness can be correlated directly with peculiarities of the film structure.