There are several issues here:
The Texas law applies to social-networking providers with 50 million users or more. That means Facebook and Twitter (and maybe YouTube).
The plaintiffs only raised point 2 above, and argued for relief based simply on the first amendment. The fifth circuit did not buy this:
In urging such sweeping relief, the platforms offer a rather odd inversion of the First Amendment. That Amendment, of course, protects every person’s right to “the freedom of speech.” But the platforms argue that buried somewhere in the person’s enumerated right to free speech lies a corporation’s unenumerated right to muzzle speech.
The implications of the platforms’ argument are staggering....
Today we reject the idea that corporations have a freewheeling First Amendment right to censor what people say. Because the district court held otherwise, we reverse its injunction and remand for further proceedings
But the idea that a publisher has a first-amendment right not to publish selected speech is in fact quite well established. The fifth circuit got around this by invoking Section 230 against the plaintiffs, arguing that since Section 230 declares that social-media companies are legally not "publishers", they lose this first-amendment publisher protection. Arguably, though, Section 230 just says that social-media companies cannot be treated as publishers for purposes of assigning liability; this does not mean that they are not publishers.
Still, the case is far from settled. Section 230 -- which generally supersedes state laws -- states flat-out that social-media companies may select what messages to carry. There is also the Takings clause, that the right to control content is an important property right, and companies may not be stripped of this right without compensation. Again, these two issues were not brought up by the plaintiffs.
From the preamble of the law itself:
(3) social media platforms function as common carriers, ....
There is no statutory basis for making common carriers out of social-media companies, although "function" does not exactly claim that. Common carriers, like the phone company, generally provide one-to-one communications. Social media is one-to-many, with the "many" possibly at the discretion of the provider. (Also, one of Trump's early actions was to remove common-carrier status from ISPs (who arguably are common carriers).)
There is an argument to be made that Twitter, with its speaker-follower mechanism, could be considered a common carrier. But Twitter also has a very real commercial interest in being allowed to moderate content.
The Supreme Court decided in the Pruneyard Mall case  that a public mall could not exclude people who wanted to collect petition signatures. Since the mall was refusing permission based on what the petition said, this can be seen as overruling a form of censorship. However, there are quite a few differences between malls and social-media sites. For one, malls are not advertising-supported. For another, the decision was based in part on details of the California State Constitution, which broadly supported free-expression in public places.Section 120.051:
Sec. 120.051. PUBLIC DISCLOSURES.
(a) A social media platform shall, in accordance with this
subchapter, publicly disclose accurate information regarding its content
management, data management, and business practices, including specific
information regarding the manner in which the social media platform:
(1) curates and targets content to users;
(2) places and promotes content, services, and products, including its own content, services, and products;
(3) moderates content;
(n) a whole bunch more
This requires that social media companies explain their decisions. This is not particularly onerous, and they all do that now to an extent, though requiring publication of the internal ranking algorithms (implicit in how the companies "curate and target" content) is controversial.
Social-media platforms will have to create "biannual transparency reports".
Sec 120.101: Social-media companies must create a complaint system, so users can complain about content taken down. The law also requires an appeals procedure. All this is quite expensive.
A social media
platform may not censor a user, a user's expression, or a user's ability to receive the expression of another person based on:
(1) the viewpoint of the user or another person;
(2) the viewpoint represented in the user's expression or another person's expression; or
(3) a user's geographic location in this state or any part of this state.
I think (3) here is supposed to mean that Facebook cannot just cut Texas off entirely.
The law does not really make clear what constitutes "censor[ing] a user"; for example, does downranking a post count? This section would apply most clearly to Twitter, where to "censor" would be to disallow all of someone's followers from receiving a tweet from the user. Facebook also has support for following someone, but the exact position of the posts in question depends on mysterious algorithms (though you can list the person you follow as a "favorite", in which case their posts should be at the top of your newsfeed). For more on Facebook following, see www.makeuseof.com/facebook-friends-followers-differences. For the official Facebook description, see www.facebook.com/help/279614732052951.
On the face of it, HB 20 does not seem to apply to newsfeed downranking.
There are rules permitting some censorship: anything illegal, for example. But things that do appear to be covered "viewpoints" include:
Note that social-media companies would be prohibited from engaging in viewpoint censorship even if the speech in question violated the site's terms of service. (I suppose the site could leave the offending post up, but kick the user off.)
All this would be a significant problem for social-media companies. First, the kind of speech they have been "censoring" is known to drive many users away. Second, it is even more effective at driving advertisers away; advertisers are extremely reluctant to have their ads appear alongside "hate speech".
A couple issues: