Not able to help on your exact question on UK- French differences but here is maybe a go that will help you.
Airline Operator Certificate is a simple certificate. The AOC is used in cooperation with Operation Specifications that outline in detail what the company is allowed to do and how. Most CAA's will issue normally a blanket AOC with vary basic details and as things change the operator will update the Ops Spec's to reflect reality.
On the other side, an AOC can have any amount of information depending on the country. I have seen them with everything from only the airline address to aircraft type and registration, type of operations (pax/cargo/mail, scheduled/charter,...), base of operations, routes allowed, dangerous goods,...
Operation Specifications are minute details of the AOC that contains all the additional information and approvals that make up the real meat of what the company is allowed to do and not do.
In the end the ICAO/JAA/EASA/FAA set a regulatory frame work that all airlines worldwide need to work within. The exact rules may be slightly different but in the end they same information must be documented meeting the international standards.