B Corps allow the board to weigh things besides shareholder value. That's a meaningful distinction.
The idea is that shareholder primacy isn't compatible with everything every corporation wants to do, so having a board that's protected from lawsuits when they put things above shareholders is a useful thing and B Corps offer that.
The board can, for example, reject a "superior" takeover bid without fear of lawsuits from shareholders pissed off they didn't get the biggest payday available. A typical C Corp's board MUST take the highest offer, and not doing so WILL get them sued. That means if GoodGuy B Corp is about to be taken over by BadGuy Inc., the GoodGuy board can say "No, they're not compatible with the public benefit mission we incorporated under so we're not going to accept their offer." That's actually really useful.
It has been a "rallying cry" but it doesn't stand up to much scrutiny of how Bluesky actually functions: an "open protocol" with one central server means little. Maybe this will change at some point in the future, and maybe it is changing, see https://blacksky.community/ . But this is not the same as Mastodon, where it's been that way for a while.
The term they use for this is "credible exit" - designing the entire protocol such that if the company itself misbehaves the affected users can leave to a separate instance without losing their relationships or data.