The difference is that if private institution that issues "toms safety certification" takes a bribe it is a very shortsighted strategy - such certificates will be very soon considered untrustworthy and general public will start to ignore them, therefore putting certificate institution out of business. Government-powered institutions can take bribes for years, or even decades without ever being punished for that, or being put out of business.

And yes, limiting government competencies is the best way to limit corruption.As one of Polish politicians put it:

"You don't fight the leeches by catching them by their tails, you fight them by draining the swamp they live in."