The vehicle routing problem with task priority and limited resources (VRPTPLR) is a generalized version of the vehicle routing problem (VRP) with multiple task priorities and insufficient vehicle capacities. The objective of this problem is to maximize the total benefits. Compared to the traditional mathematical analysis methods, the pointer neural network proposed in this paper continuously learns the mapping relationship between input nodes and output decision schemes based on the actual distribution conditions. In addition, a global attention mechanism is adopted in the neural network to improve the convergence rate and results. To verify the effectiveness of the method, we model the VRPTPLR and compare the results with those of a genetic algorithm. The parameter sensitivity of each algorithm is assessed using different datasets. Then, comparison experiments with the two algorithms employing optimal parameter configurations are performed for the validation sets, which are generated at different instance scales. It is found that the solution time of the pointer neural network is much shorter than that of the genetic algorithm and that the proposed method provides better solutions for large-scale instances.