It’s not so much about getting paid by NYT, but rather about NYT going through every word they paid for, and deciding whether to publish it or not after all. Facebook and Twitter do not do that: while they do have some very basic automated filters, they are extremely hands off with the large majority of the content on their platform, only reviewing maybe a single digit percentage of published stuff, and only after a fact. Because of this, they claim that they shouldn’t be legally liable on content published on their platform, as they weren’t making any decisions whether to publish it or not. This is the major difference with NYT: NYT does not claim that it’s just random writers publishing in it, and they don’t pay much attention to what is inside.