This paper presents an adaptive neural network control sche-me with quantitative prescribed performance for an autonomous underwater vehicle (AUV) equipped with forward/aft vertical thrusters. A backstepping-based controller is designed using error transformation, ensuring predefined constraints on both transient overshoot and convergence time of tracking errors. To handle model uncertainties and external disturbances, an online approximator based on a radial basis function neural network (RBF-NN) is employed for real-time compensation. Additionally, the dead-zone and saturation nonlinearities inherent in thruster inputs are addressed by reformulating them into a continuously differentiable smooth model via a hyperbolic tangent function. Simulations validate the proposed method under the scenarios with parametric uncertainties, external disturbances, and thruster input nonlinearities with dead-zone and saturation.