The FAA and EUROCONTROL jointly published a document in 2013 that provides a comparison of air traffic management starting in section 2.
Most differences have to do with the fact that the EU is comprised of multiple countries, while in the US, the FAA has full authority.
Organization:
The US system is operated by one single service provider using the same tools and equipment, communication processes and a common set of rules and procedures ... [while] the European system is much more fragmented and ANSPs are still largely organised by State boundaries.
Route/airspace management (including special use airspace):
[T]he Federal Aviation Administration (FAA) is responsible for airspace management and route design[;] in the amalgamated European ATM system, airspace design remains the prerogative of the individual States.