IIRC, both do the same thing but are managed very differently. OPM is funded by charity/the public BUT encompasses the world whereas in MHA its a government body that only encompasses a country. Both require licences for heroes to be officially recognised.
OPM has weird issues with the hero association appearing incompetent (but justified with how bat shit crazy things have become). They're also ran by the top brass heroes. Was founded fairly recently which sort of excuses them, but still kind of annoying seeing their base get wrecked over and over, infighting, egos vs egos when their world is on the line, etc.
MHA is much more established and is solely ran by people without quirks to reduce corruption/conflict in interests between heroes and regular citizens.
OPM encourages heroes to actively save people, considering they're publicly funded. A bad relationship/failure to help = bad return and distrust. Also, OPM has a rather simple test to become a hero.
MHA actually fosters heroes. In that regard it's significantly better since aspiring heroes will be given a chance to grow before they're sent out, rather than grab a licence straight away and potentially get bodied in their first fight.
As much as I love OPM's HA for some reason (because it looks cool), MHA's commission is much better for heroes in my opinion.
Please dismiss this if I'm totally wrong though. I could be misremembering things.