There is quite a difference: The station isn’t paid for by taxes from the government, but the station has the right to collect its own taxes, ensuring it can act independently.
Side note: as someone who pays that fee, I would never think of myself as a 'customer' of public broadcasting, since it's a public service -- just like I wouldn't think of myself as a 'customer' of hospitals, schools, communal waste disposal, the fire department or law enforcement.
That may be true, but it's irrelevant to my point, which was that the Rundfunkbeitrag is a matter of state (Land) law, and thus even further away from the hands of the federal foreign ministry.