This paper presents two controllers that address the negative effects of external disturbances, parameter variations, and random noise on trajectory tracking for robot manipulators. These controllers were designed for handling variable-mass loads, uncertainties, and disturbances in the robot’s dynamic equations. In order to achieve an accurate trajectory tracking and an improved disturbance rejection, Terminal Sliding Mode Control (TSMC) was employed in combination with a Neural Network (NN) with radial basis functions that modelled the unidentified system characteristics. The employed control algorithms, namely the Neural Network-Based Terminal Sliding Mode Control (NNBTSMC) and the Neural Network-Based Backstepping Terminal Sliding Mode Control (NNBBTSMC), were optimized using genetic algorithms in order to determine the optimal controller coefficients. Simulation tests were conducted on a two-link robot manipulator, and the effectiveness of the proposed controllers was demonstrated. Additionally, a comparison was made between the analysed controllers regarding their trajectory tracking performance and control inputs for the case when the manipulator payload changed over time. The simulation results showed that the NNBBTSMCWGA outperformed the traditional SMC and NNTSMC by exhibiting a higher robustness, and reduced chattering effects on the control inputs.