Continuum robot is a type of robot with continuous flexible skeleton, which plays an important role in production and life. At present, the position and orientation control (POC) scheme is not widely implemented in continuum robots, and it does not pay attention to the dynamic tracking error of the end-effector orientation, which has obvious limitations for completing the task. In view of the shortcomings of the existing research, this paper proposes a hybrid neural network-based algorithm for controlling the position and orientation of a continuum robot. In this algorithm, the POC scheme is applied to the continuum robot. Based on the kinematics equation of the continuum robot, the corresponding dynamic neural network (DNN) and the dynamic tracking algorithm of the end-effector orientation are obtained by theoretical derivation. The simulation results show that the proposed scheme reduce the position error of the end-effector of the continuum robot and improve operability. Specifically, the tracking error of the Euler angle converges quickly, and the ideal Euler angle is basically consistent with the actual Euler angle. The comparison between the experimental results and the existing control methods verifies the feasibility and advancement of the scheme.