Members of the public are expected to meet one standard. There are additional -higher- standards placed upon people in public office.

I don't understand how a law that applies to a certain class of people, but not everybody, is not treating that certain class of people differently.

The law creates the higher standard. Behaviour that's perfectly legal for most people becomes illegal for people in public office because we expect higher standards from those people and we expect these higher standards so strongly we created a law to enforce it.