Federal agencies can apply alternate standards if they decide general OSHA rules don't work for them; while this requires DoL sign-off, IIRC, this option is not available to individual private employers.

> What rules apply to the private sector that don't apply to the public sector?

Aside from the above, another example is that a variety of regulations under FLSA, including standards for exemption, do not apply to federal employees when adopted by the Labor Department; OPM sometimes adopts conforming regulations that apply similar standards to the federal civilian workforce and sometimes does not.

Similarly, states often limit the applicability of their own labor laws to themselves, though they are of course bound by federal laws to the extent that they aren't either specifically exempted or Constitutionally immune.

And all Workman's compensation claims go through a government agency (Department of Labor).

With zero ability to sue or claim damages like you would have in other businesses.

If you get hurt via some hilariously OSHA-illegal workplace incident, the most you can hope for is a schedule reward of some number of your work weeks plus forced retirement, and most people have to fight to get that using their own resources and legal help. All the while being treated as a criminal or a draft dodger.

So in the end, an OSHA-unsafe environment has no repercussions to governmental organizations ($$ as normal or firing).