Interfacial properties of binary azeotropic mixtures of Lennard-Jones truncated and shifted fluids were studied by molecular dynamics (MD) simulation and density gradient theory (DGT) in combination with an equation of state. Three binary mixtures were investigated, which differ in the energetic cross interaction parameter that yields different types of azeotropic behavior. This study covers a wide temperature and composition range. Mixture A exhibits a heteroazeotrope at low temperatures, which changes to a low-boiling azeotrope at high temperatures, mixture B exhibits a low-boiling azeotrope, and mixture C exhibits a high-boiling azeotrope. The phase behavior and fluid interfacial properties as well as their relation were studied. Vapor-liquid, liquid-liquid, and vapor-liquid-liquid equilibria and interfaces were considered. Density profiles, the surface tension, the interfacial thickness, as well as the relative adsorption and enrichment of the components at the interface were studied. The results obtained from the two independent methods (MD and DGT) are overall in good agreement. The results provide insights into the relation of the phase behavior, particularly the azeotropic behavior, of simple fluid mixtures and the corresponding interfacial properties. Strong enrichment was found for the mixture with a heteroazeotrope in the vicinity of the three-phase equilibrium, which is related to a wetting transition.