We study the thermophoretic motion of water nanodroplets confined inside carbon nanotubes using molecular dynamics simulations. We find that the nanodroplets move in the direction opposite the imposed thermal gradient with a terminal velocity that is linearly proportional to the gradient. The translational motion is associated with a solid body rotation of the water nanodroplet coinciding with the helical symmetry of the carbon nanotube. The thermal diffusion displays a weak dependence on the wetting of the water-carbon nanotube interface. We introduce the use of the moment scaling spectrum (MSS) in order to determine the characteristics of the motion of the nanoparticles inside the carbon nanotube. The MSS indicates that affinity of the nanodroplet with the walls of the carbon nanotubes is important for the isothermal diffusion and hence for the Soret coefficient of the system.