I am not a lawyer, but my understanding is the truth of legal systems is that they are a gray area of interpretation and precedent. In a strict definition of copyright, copying a file from a server into the cache in your ram is a copy that theoretically could be ruled an unauthorized copy. This being obviously a ridiculous idea, but I believe if you streamed content from a website this is the only way they could fine you for copyright violation. Generally with the networks you mentioned, the distribution is prohibited, and most being peer to peer systems they cite you for that.

The more charitable take on why they are not the same is; anything on the open web is assumed to give the right to copy to a closed 'local' system and use for your own system, as that is fundamental to web browsing. Further in the same way you can make a local copy of a movie to cut it up and use in a review, and that is fair use as a transformative work, you can make copy of the open web content and make a transformative work for it.

At least that's how I would argue it, if I was paid to be OpenAI's legal puppet. All of that is to tiptoe around the reality that the legal system is a tool of the rich. So the law becomes firm for the purpose of piracy of a movie, but long and difficult for the purpose of a large companies profits.