The two terms are almost the same thing, in general we use the word "deep" model to talk about neural network based model, in fact it should describe neural networks that are "deep", but there is no real size from which we consider that a neural network is a "deep" or "shallow". As a conclusion, in your case reinforcement neural network is the same as deep reinforcement learning.
Thank you Mohamed-Mourad Lafifi, for the interesting papers. I will add Article Playing Atari with Deep Reinforcement Learning
, which is a classical paper for deep RL, I also add this interessting link from DeepMind, about AlphaStar, one of the biggest achievments in the world of RL in the last few years. https://deepmind.com/blog/alphastar-mastering-real-time-strategy-game-starcraft-ii/. I hope you enjoy reading them.