A linear time-invariant (LTI) system is indeed completely described by its frequency response. Note that the frequency response is the Fourier transform of the impulse response, which also completely describes the system. So if the Fourier transform of the impulse response exists, then the resulting frequency response must represent a complete characterization of the system.
In many cases both the Fourier transform and the Laplace transform of the impulse response exist, and, consequently, both contain the same information. In these cases, the Laplace transform (transfer function) does not add any information that isn't already represented by the Fourier transform (frequency response).