This paper focuses on the adaptive trajectory tracking control for a remotely operated vehicle (ROV) with an unknown dynamic model and the unmeasured states. Unlike most previous trajectory tracking control approaches, in this paper, the velocity states and the angular velocity states in the body-fixed frame are unmeasured, and the thrust model is inaccurate. Obviously, it is more in line with the actual ROV systems. Since the dynamic model is unknown, a new local recurrent neural network (local RNN) structure with fast learning speed is proposed for online identification. To estimate the unmeasured states, an adaptive terminal sliding-mode state observer based on the local RNN is proposed, so that the finite-time convergence of the trajectory tracking error can be guaranteed. Considering the problem of inaccurate thrust model, an adaptive scale factor is introduced into thrust model, and the thruster control signal is considered as the input of the trajectory tracking system directly. Based on the local RNN output, the adaptive scale factor, and the state estimation values, an adaptive trajectory tracking control law is constructed. The stability of the trajectory tracking control system is analyzed by the Lyapunov theorem. The effectiveness of the proposed control scheme is illustrated by simulations.