This paper transforms the discrete Langrangian formulation of spatial flexible tether systems to Hamiltonian formulism. The resulting Hamiltonian canonical equations are solved by Symplectic difference scheme. The application of Symplectic difference method for solution of nonlinear tether dynamic problems ensures conservation of system energy, momentum and volume. Two numerical examples are conducted to validate the proposed method, one is the free swing pendulum system and the other one is the three-dimensional circular towed system. The simulation results are compared with theoretical and the existing numerical results. The comparisons demonstrate the proposed Symplectic difference integrator for the Hamiltonian nodal position finite element method is numerically accurate and efficient to predict the dynamic response of spatial flexible tether systems.