Surface spin waves in thin permalloy films are studied by means of propagative-spin-wave spectroscopy. We observe a systematic difference of up to several tens of MHz when comparing the frequencies of counterpropagating waves. This frequency nonreciprocity effect is modeled using an analytical dipole-exchange theory that considers the mutual influence of nonreciprocal spin wave modal profiles and differences in magnetic anisotropies at the two film surfaces. At moderate film thickness (20 nm and below), the frequency nonreciprocity scales linearly with the wave vector and quadratically with the thickness, whereas a more complex nonmonotonic behavior is observed at larger thickness. Our paper suggests that surface wave frequency nonreciprocity can be used as an accurate spectroscopic probe of magnetic asymmetries in thin ferromagnetic films.