Omnidirectional mobile robots have gained a lot of attention in recent years due to their maneuverability capabilities. However, ensuring accurate trajectory tracking with this class of robots is still challenging control system designers. In this work, a novel intelligent controller is introduced for accurate trajectory tracking of omnidirectional robots subject to unstructured uncertainties. An adaptive neural network is adopted within a Lyapunov-based nonlinear control scheme to deal with frictional forces and other unmodeled dynamics or external disturbances that may occur. Online learning, rather than supervised offline training, is employed to allow the robot to learn on its own how to compensate for uncertainties and disturbances by interacting with the environment. The adoption of a combined error signal as the single input in the neural network significantly reduces the computational complexity of the disturbance compensation scheme and enables the resulting intelligent controller to be implemented in the embedded hardware of mobile robots. The boundedness and convergence properties of the proposed control scheme are proved by means of a Lyapunov-like stability analysis. The effectiveness of the proposed intelligent controller is numerically evaluated and experimentally validated using a omnidirectional mobile robot. The comparative analyses of the obtained results show that the adoption of an intelligent compensation scheme based on adaptive neural networks allows reductions of more than 95% in the tracking error, thus guaranteeing an accurate tracking and confirming the great superiority of the proposed control strategy.