In this paper, continuous and event-sampled approximate optimal distributed control schemes for an interconnected system, with nonlinear subsystem dynamics and strong interconnections, are presented. The control design problem for the interconnected system is reformulated as an N-player cooperative nonzero-sum differential game wherein the control policy of each subsystem is treated as a player in the game. The Nash solution of this game is used to design the control policy for each subsystem to optimize the performance of the interconnected system. Approximate dynamic programming (ADP), with critic neural networks, is utilized to approximate the solutions of the coupled Hamilton-Jacobi equations, for continuous and event-sampled control implementation. Event-sampling conditions are designed to asynchronously orchestrate the sampling and transmission at each subsystem. Finally, simulation results are included to substantiate the theoretical claims.