The hydrophobic effect, i.e., the poor solvation of nonpolar parts of molecules, plays a key role in protein folding and more generally for molecular self-assembly and aggregation in aqueous media. The perturbation of the water structure accounts for many aspects of protein hydrophobicity. However, to what extent the dispersion interaction between molecular entities themselves contributes has remained unclear. This is so because in peptide folding interactions and structural changes occur on all length scales and make disentangling various contributions impossible. We address this issue both experimentally and theoretically by looking at the force necessary to peel a mildly hydrophobic single peptide molecule from a flat hydrophobic diamond surface in the presence of water. This setup avoids problems caused by bubble adsorption, cavitation, and slow equilibration that complicate the much-studied geometry with two macroscopic surfaces. Using atomic-force spectroscopy, we determine the mean desorption force of a single spider-silk peptide chain as F = 58 +/- 8 pN, which corresponds to a desorption free energy of approximately 5 k(B)T per amino acid. Our all-atomistic molecular dynamics simulation including explicit water correspondingly yields the desorption force F = 54 +/- 15 pN. This observation demonstrates that standard nonpolarizable force fields used in classical simulations are capable of resolving the fine details of the hydrophobic attraction of peptides. The analysis of the involved energetics shows that water-structure effects and dispersive interactions give contributions of comparable magnitude that largely cancel out. It follows that the correct modeling of peptide hydrophobicity must take the intimate coupling of solvation and dispersive effects into account.