Not being privy to the way the French classify their NOTAM I can't give you a definitive answer, I was only passing on what I had been told by UK AIS.
ICAO Annex 15 allows for NOTAM to be grouped in Series. Recipients can then decide whether or not they wish to be on the distribution list for particular Series.
The NOTAM Series for the UK are listed in table 3.1.1 in GEN 3.1 of the UK AIP. The French AIP is not yet available on-line so I can't point you at it.
What you get from a briefing service or from UK AIS are not NOTAM, they are Pre-Flight Information Bulletins (PIB's) derived from NOTAM. The briefing service will decide which series are included in the data they use to produce the PIB. For example Series Q (Military Series (Sovereign Bases)) would not be much use in a brief for the UK FIR's.
Examples of things that could be in French domestic NOTAM but not in their International series:-
Regulatory notices relating to licensing of aircraft or pilots.
Facilities at airfields not available for International use.
Obstacles in the approach path at airfields not available for International use.
The EAD is slowly creeping forward. I couldn't get in at all at first, now I can, so they are obviously slowly sorting out the software. The prospect of having centralised access to European AIP's and AIC's is tempting, particularly as and when the electronic AIP becomes a reality (this will present the AIP in a fully searchable electronic format with hyperlinks rather than as images of paper pages). You would be able to download plates as required rather than subscribe to larege quantities of paper and their amendments, most of which you probably do not use.
Time will tell if it becomes a success.
Mike