We examine the thermodynamics of the liquid-vapor interface by direct calculation of the surface entropy, enthalpy, and free energy from extensive molecular dynamics simulations using the two-phase thermodynamics (2PT) method. Results for water, acetonitrile, cyclohexane, dimethyl sulfoxide, hexanol, N-methyl acetamide, and toluene are presented. We validate our approach by predicting the interfacial surface tensions (IFT-excess surface free energy per unit area) in excellent agreement with the mechanical calculations using Kirkwood-Buff theory. Additionally, we evaluate the temperature dependence of the IFT of water as described by the TIP4P/2005, SPC/Ew, TIP3P, and mW classical water models. We find that the TIP4P/2005 and SPC/Ew water models do a reasonable job of describing the interfacial thermodynamics; however, the TIP3P and mW are quite poor. We find that the underprediction of the experimental IFT at 298 K by these water models results from understructured surface molecules whose binding energies are too weak. Finally, we performed depth profiles of the interfacial thermodynamics which revealed long tails that extend far into what would be considered bulk from standard Gibbs theory. In fact, we find a nonmonotonic interfacial free energy profile for water, a unique feature that could have important consequences for the absorption of ions and other small molecules.