What are the differences between approximation of AIXI and deep reinforcement learning
It's a bit of an apples and oranges comparison. AIXI is a proposal for how generalized learning should be structured, while neural networks are a specific learning implementation.

That being said, you probably wouldn't want to use an AIXI implementation for a known task such as game AI. Game AI is a specific learning task, so it's better and easier to use an AI specifically tailored to that task. The AlphaGo AI, for example, combined a deep learning neural network for finding interesting looking moves with a positional analysis engine for figuring out the strength of those moves. That sort of architecture works great for learning to play Go, but it's not general.