The fundamental role of hydration water (also called interfacial water) is widely recognized in protein flexibility, especially in the existence of the so-called protein "dynamical transition" at around 220 K. In the present study, we take advantage of perdeuterated C-phycocyanin (CPC) and elastic incoherent neutron scattering (EINS) to distinguish between protein dynamics and interfacial water dynamics. Powders of hydrogenated (hCPC) and perdeuterated (dCPC) CPC protein have been hydrated, respectively, with D(2)O or H(2)O and measured by EINS to separately probe protein dynamics (hCPC/D(2)O) and water dynamics (dCPC/H(2)O) at different time- and length-scales. We find that "fast" (<20 ps) local mean-square displacements (MSD) of both protein and interfacial water coincide all along the temperature range, with the same dynamical transition temperature at ~220 K. On higher resolution (<400 ps), two different types of motions can be separated: (i) localized motions with the same amplitude for CPC and hydration water and two transitions at ~170 and ~240 K for both; (ii) large scale fluctuations exhibiting for both water molecules and CPC protein a single transition at ~240 K, with a significantly higher amplitude for the interfacial water than for CPC. Moreover, by comparing these motions with bulk water MSD measured under the same conditions, we show no coupling between bulk water dynamics and protein dynamics all along the temperature range. These results show that interfacial water is the main "driving force" governing both local and large scale motions in proteins.