As stated in this review article:
Mechanical modes are long compared to the interatomic spacing. It is natural to make the distinction between nanomechanical modes and phonons: The former are lowfrequency, long-wavelength modes strongly affected by the boundary conditions of the nanodevice, whereas the latter are vibrational modes with wavelengths much smaller than typical device dimensions. Phonons are relatively unaffected by the geometry of the resonator and [...] are essentially identical in nature to phonons in an infinite medium
My feeling is, however, that "phonon" is used in all sorts of different contexts, is usually interchanged with mechanical mode, and is even used where there is no vibrating solid (e.g. in harmonic oscillators). Could you help me clarify a bit the difference and the nomenclature?