It's the fact that the state has, and is currently, proping it up with total authority as a public square. Through funding, through exclusion from regulation, through back-room deals with other industries at the government's behest.

Twitter.com shouldn't be the public square, but the government doesn't want decentralized communication. I'm happy letting Twitter do whatever it wants if it were a private company that was exclusive to itself and not operating as the extension of discussion from the state. That, however, requires ripping the fingernails out of the government. Since we can't do that yet, we have to recognize that a chilling effect on speech is very obviously taking place when the government is trying to engage in viewpoint discrimination.