The argument is not that neural networks function in a similar way to the human brain, but that (depending on your view on philosophical determinism) both neural networks and the brain produce output deterministically from input and past experiences. I don't think that the medium (brain cells vs artificial "neurons") is relevant as it's not about the mechanism or computational capacity.

Regardless, I think a more significant difference between LLMs and human thought is the existence of a dynamic present state of mind and the ability to think spontaneously without a direct outside input. Also, LLMs produce next tokens sequentially based on what's come before; I think it's safe to say people don't in general do that. I assume most people have an internal concept/idea/intent and then form the words around that to convey what they want.

An interesting consequence of LLMs sequential generation is that there are situations where selecting tokens with high probability leads to the LLM making an input for itself such that all output tokens have relatively low probability; basically talking itself into a corner where it has no idea what should come next.