The theory of interfacial tensions, ITC theory, developed over the last half-century from contact angle and wetting studies, has proven to be reliable in many fields of physical and biophysical chemistry. However, interfacial tensions for curved solid surfaces in liquids, estimated in various ways from the Kelvin equation (itself, of course, a very successful theory), give very different results from those estimated by ITC theory. That is, a clear distinction must be made between the Kelvin equation parameter (KEP) of classical freezing theory and the contact angle parameter (CAP) of ITC theory. The difference between KEPs and CAPs is to some degree caused by the incorrect application of addition rules, Antonow’s rule, for example, but a more profound discrepancy remains even when correct addition rules are used. One source of a discrepancy in. measures of solid-vapor surface tension has been known since Gibbs, but has often not been acted upon, and does not appear to have been routinely quantified. Discrepancies in solid-liquid interfacial tensions have not, to my knowledge, been identified and discussed quantitatively in such detail before. In this paper, solid-liquid CAPs, from ITC theory, and KEPs, from freezing theory, are given for water and for naphthalene. Naphthalene is apolar and so is independent of certain aspects of ITC theory. Both substances show a large discrepancy between their KEP and their CAP. The effects of the discrepancy on the freezing of water ice are described. Several possible reasons for the discrepancy are discussed. Of the reasons for the discrepancy that have been identified, order-of-magnitude quatification suggests that only strain in the solid lattice or gross systematic differences between macroscopic and molecular level contact angles are potentially large enough to account for it. If there is a lattice strain component in KEPs and the capillarity approximation is to be retained in freezing studies, there may be an additional stochastic effect in freezing, which will determine the magnitude of the interfacial tension of nucleating clusters and which will be very difficult to calculate from first principles.