Transversely stratified jets are observed in many classes of astrophysical objects, ranging from young stellar objects, mu-quasars, to active galactic nuclei and even in gamma-ray bursts. Theoretical arguments support this transverse stratification of jets with two components induced by intrinsic features of the central engine (accretion disk + black hole). In fact, according to the observations and theoretical models, a typical jet has an inner fast low density jet, surrounded by a slower, denser, extended jet. We elaborate on this model and investigate for the first time this two-component jet evolution with very high resolution in 3D. We demonstrate that two-component jets with a high kinetic energy flux contribution from the inner jet are subject to the development of a relativistically enhanced, rotation-induced Rayleigh-Taylor type non-axisymmetric instability. This instability induces-strong mixing between both components, decelerating the inner jet and leading to overall jet decollimation. This novel scenario of sudden jet deceleration and decollimation can explain the radio source Fanaroff-Riley dichotomy as a consequence of the efficiency of the central engine in launching the inner jet component versus the outer jet component. We infer that the FRII/FRI transition, interpreted in our two-component jet scenario, occurs when the relative kinetic energy flux of the inner to the outer jet exceeds a critical ratio.