There are some exceptions — for example, a flight instructor is normally allowed to fly for money in a private aircraft owned by the student — but the above requirements hold for most flights where money changes hands.
Typically, a commercial license or registration requires higher standards than a private one. For example, a commercial pilot may have to demonstrate more maneuvers to a higher standard, and may need to pass more frequent medical examinations. A commercially-registered plane may require more frequent or more extensive maintenance.
It is the purpose of the flight, not the type of aircraft or pilot, that determines whether the flight is commercial. For example, a two-seat Cessna 150 towing a banner for money would be a commercial flight, while a large jet flown by its owners for a private vacation would not be, even if the pilots were commercially licensed and the jet were commercially registered.