My opinion of the B Corp designation, which has been heavily colored by founding + running a nonprofit full-time, is that it's mostly a marketing tool, and otherwise doesn't carry much weight. A common context I come across them in is offering services to nonprofits (ex [3] [4]), which they seem to do at approximately market rates in most cases.
In some ways (but certainly not all), I think the 501c3 process/designation puts stronger guardrails in place to make sure an organization isn't doing anything particularly terrible, by 1) limiting the financial upside to doing "bad" things, 2) removing the tax-exempt status if the org deviates too far from their stated mission (filed in a 1023 with the IRS) or receives too much money from the wrong places, and 3) forcing a bit of transparency by publishing 990s (the nonprofit yearly tax filing).
[1] https://www.bcorporation.net/en-us/find-a-b-corp/company/nes...
[2] https://en.wikipedia.org/wiki/Nestl%C3%A9#Controversies_and_...