The key difference is that the government never touches the money anywhere between citizen and broadcaster, to avoid unwarranted influence. I assume that people who know the British term know that, as the German one is surely modeled after the BBC. I'm writing for other readers.
The downside is that it's a per head (or per household) sum, not coupled to income like taxes would be. This is usually explained away by the fee being separate from the state, but the reality is that Germany actually has it all implemented, in the form of the opt-in "church tax" coupled to taxable income just like regular tax. Handled by the tax office, but not going too government coffers. Would be so easy to extend the implementation to public broadcasting, because you don't pay to consume the media, you pay to live in an environment that is not dominated by profit-driven broadcasting media. There are many negative things to say about our public broadcasting, but when I look at other countries that don't have strong public broadcasting, it's so much the lesser evil, totally worth the fee.
(personally, I'd love to see that "church tax implementation" opened up to all kinds of opt-in membership organisations that would see value in income-coupled membership fees, I believe that a lot of good things could work that way, with people of all income levels enjoying an objectively fair way of contributing)