In this paper, a novel fast nonsingular integral terminal sliding mode controller based on an adaptive neural network (ANN-FNITSMC) is proposed for the trajectory tracking control of cable-driven continuum robots (CDCRs) in complex underwater environments with uncertainties. First, a novel fast nonsingular integral terminal sliding mode control (FNITSMC) is designed to solve the chattering and singularity problems of the conventional terminal sliding mode control (TSMC). Second, an adaptive neural network (ANN) based on a radial basis function (RBF) is established to derive the uncertainties and compensate for the control input of CDCRs, enabling high-stable accuracy and strong robustness trajectory tracking in complex underwater environments. Simulation results are presented to demonstrate the high accuracy and strong robustness of the ANN-FNITSMC. Finally, the high accuracy, high stability, and strong robustness of the proposed trajectory tracking strategy are verified through an underwater experiment platform.