The propagation of nonlinear nonaxisymmetric waves along a magnetic tube in an incompressible plasma embedded in a magnetic-free plasma is studied. The plasma and magnetic parameters in the tube core as well as plasma parameters in the external plasma are constant. Between the tube core and the magnetic-free plasma there is a thin annulus where the Alfvén speed monotonically decreases to zero. In this annulus there is a cylindrical surface where the phase speed of the global wave matches the local Alfvén speed. In the vicinity of this surface there is an efficient conversion of the global wave energy in the energy of local Alfvén waves. This results in the resonant absorption of the global wave and, as a consequence, in the global wave damping. The wave amplitude is assumed to be small and used as a small parameter in the singular perturbation method that is used to derive the nonlinear governing equation for nonaxisymmetric waves. This equation accounts both for nonlinearity and wave damping due to resonant absorption. A particular class of solutions of this equation in the form of helical waves is studied numerically. The main result obtained in this study is that nonlinearity accelerates the wave damping. It also distorts the shape of the tube boundary due to nonlinear generation of fluting modes.