The definition of the Hamiltonian and mean magnetization of XXZ open and closed spin chain of Heisenberg model in one-dimension is given, then the ground state, excited state energy and mean magnetization of the XXZ Heisenberg model of the open and closed spin chain in one dimension are calculated under different parameters and different configuration ［N,k］. At last, the correlation degree and phase transition of XXZ open and closed spin chain of Heisenberg model in one-dimension are analyzed comparatively according to the calculation results. Results show that the correlation degree is embodied very well when the energy difference between the ground state and the excited state change along with the parameter or configuration ［N,k］. The phase transition point of the system can be found according to the inflection point that the ground state energy changes along with the change of the parameter and the degeneracy that the energy difference between the excited state and the ground state changes with the parameter. It is also shown that the calculation results of the open and closed spin chain are same roughly, however, the calculation effect of the closed chain model is more intuitive and obvious than that of the open chain model. The correlation properties and phase transition of the XXZ open and closed spin chain of Heisenberg model in one-dimension are compared, which provides an important reference for further understanding the quantum correlation properties of Heisenberg model.