What are the advantages of neural cryptography over Diffie-Hellman algorithm? I am not sure whether such a question is accepted here, or if this is the correct place to ask. 

I have done some research on this topic, however most of the information was way too complex for me to understand. I know that the most used protocol for key exchange between two parties A and B, in practice is Diffie-Hellman protocol.

I have read here, and understood the basic working protocol for neural cryptography but do not see any clear advantages over the Diffie-Helman algorithm.

What are the advantages (if any) of neural cryptography over the Diffie-Hellman algorithm?
I am not an expert either, but from what I gather the intent is to create a key exchange mechanism that is not based on any number theory problem. The concern is that advances in computing such as quantum computing could make the existing one-way functions such as prime multiplication effectively a two-way function. The authors believe that the neural exchange process will not be as vulnerable to advances in computing.

However the authors also acknowledge that their initial implementation is vulnerable to attacks using probabilistic analysis or genetic algorithms.

So at the moment, there is no advantage. The research is based on creating a future advantage, assuming that they can create a more secure implementation, and that future attacks do in fact become more sophisticated in the way that we expect.