Random anecdote about the difference: I (in Germany) can't really remember seeing anything about what ECDC said or thinks in the past months in the media, only about country-level (and lower) authorities. Whereas US-CDC seems quoted quite a lot in US media.
Oh for sure, the CDC is certainly more talked about by the media, but the power and structure of the agencies is fairly similar.
It's only a matter of time until EU's agencies impose supremacy on its Member States. The EU supreme court is already starting to penalize member States for violating policies[1].
ECDC has primarily the "power" to coordinate data exchange between member states, and afaik no authority to force lower-level health authorities or healthcare providers to do anything, unless member states were to explicitly adopt their suggestions as binding in local law.
The mess that's going on with ECB is an IMHO quite special situation (which also is an odd example for "already starting to penalize for violating policies". There is no "starting to", obviously EU law has enforcement tools - although the court itself can't use them)
