The theory of almost periodic functions is used to investigate spectral properties of Schrodinger operators on metric graphs, also known as quantum graphs. In particular we prove that two Schrodinger operators may have asymptotically close spectra if and only if the corresponding reference Laplacians are isospectral. Our result implies that a Schrodinger operator is isospectral to the standard Laplacian on a may be different metric graph only if the potential is identically equal to zero.