We report on original nonlinear spectral hole-burning experiments in single wall carbon nanotubes that bring evidence of pure dephasing induced by exciton-exciton scattering. We show that the collision-induced broadening in carbon nanotubes is controlled by exciton-exciton scattering as for Wannier excitons in inorganic semiconductors, while the population relaxation is driven by exciton-exciton annihilation as for Frenkel excitons in organic materials. We demonstrate that this singular behavior originates from the intrinsic one-dimensionality of excitons in carbon nanotubes, which display unique hybrid features of organic and inorganic systems.