Basically, the difference is whether the US government can compel private organizations to give a platform to someone.
The idea of "incorporating" the Bill of Rights against states and cities is a result of the Due Process clause of the 14th amendment. (For example, I wouldn't be surprised if the Heller decision would eventually result in US cities like NYC, SF and LA being unable to require gun licenses anymore, after someone sues them like in MacDonald vs Chicago).
That said, Facebook and Twitter are not publicly owned, they are "private" corporations (which are publicly traded). The problem isn't their specific decisions, but that they are too big.
The Sherman Antitrust act actually gave a lot of teeth to the idea that the public government can break up businesses if they abuse their monopoly power (and supported even by Republican presidents like Teddy Roosevelt). Ma Bell. Standard Oil. Etc. Today, people look at Facebook and Google as targets for breaking them up into competing companies or unbundling their departments from favoring one another.
As a left-libertarian, I would instead like to subsidize open-source software to disrupt these Big Tech monopolies from the ground up. And it's a subtle point, but I would like for the State to simply pass laws explicitly declining to enforce property rights and other rights of entities that "control too much". Landlords who own too many houses, or banks that merge together into large conglomerates will be put on notice that the public system will not enforce their rights against members of the public, past a certain point. It doesn't serve society for private property to have no limits, similarly to how you can't scale up Roman era ships and still have them seaworthy.
I recently gave an interview at Glasgow Caledonian University about the economics of free speech, where I go into detail about the problems and solutions in all the aspects around Capitalism and Free Speech, from money in politics, to deplatforming, to Big Tech filter bubbles, to clickbait news, to the Sinclair television chorus. It contains a lot of critiques of Capitalism and Competition as the basis for free speech.
If anyone has about 30 minutes to listen through it, I would love your feedback: https://youtu.be/M8HbvC6vqIY