A variational model for epitaxially-strained thin films deposited on substrates is derived by Γ-convergence from the so-called transition-layer model available in the literature. The regularity of energy-minimal film profiles is studied by establishing the internal-ball condition and by implementing some arguments from transmission problems. The possibility of different elastic properties between the film and the substrate is included in the analysis, as well as the surface tensions of all three involved interfaces: film/gas, substrate/gas, and film/substrate. The results relate to both the Stranski-Krastanow and the Volmer-Weber modes. Moreover, geometrical conditions are provided for the optimal wetting angle, i.e., the angle formed at the contact points between films and substrates. In particular, the Young-Dupr\`e law is shown to hold, yielding what appears to be the first analytical validation of such law in the context of Continuum Mechanics for a thin-film model.