Mak, an answer and website links here, then more details by email to you (and anyone else wishing to read it all).
AOPA principally caters for GA pilots who wish to fly without building their own, or entering competitions, or setting out to break records. It also has corporate members, mainly flying schools. On the airfield front, it assists operators who are members with planning issues on their specific aerodromes. It supports planning arguments with Government and local authorities etc. via GAAC, of which it is a member organisation. See
http://www.aopa.co.uk.
PFA principally caters for power pilots who wish to build and/or maintain their own aircraft. There is also a some competition element, and it is a conduit for setting out to break certain records, for which it is necessary to belong to an association which is a member of the Royal Aero Club, which PFA is. On the airfield front, it supports planning arguments with Government and local authorities etc. via GAAC, of which it is a member organisation. See
http://www.pfa.org.uk.
(The Royal Aero Club is the UK federation of air sport controlling bodies, all of which have some sporting, competitive, and/or record element. It is a club of Associations. Other RAeC members include BGA, BBAC, BMAA etc. covering gliding, ballooning, microlites, etc.. Individuals can join too, but they are not a primary focus of RAeC. See
http://www.royalaeroclub.org/raechome.htm.)
GAAC was set up by AOPA and PFA initially, then joined by around 80 other organisations including almost all those in membership of RAeC, all concerned at the general planning issues affecting aerodromes. It is almost the only place where AOPA and the sporting aviation bodies come together in common cause. Individuals can support it too, and their financial contributions all help. See
http://www.gaac.co.uk.
Each of the separate associations has been found necessary in the past because they cater for widely different needs, have different specialities in knowledge and support for their pilots, and sometimes have different agendas from each other on certain topics. The question of GAAC taking on a larger common role was raised at its last AGM, and it was clear that only on the planning front is there enough common ground for GAAC to speak for all. Also, it is all GAAC has funds for. It is the creature of its creators and financial supporters, which is largely BGA, AOPA, PFA and smaller contributions from the other associations, plus some indiviual members and many small groups such as PFA struts. Those who pay call the tune, and general (not site-specific) planning issues are what they pay it to tackle.
BGA and BMAA provide specific site planning help to various extents for their specialised flying sites, as AOPA does for its aerodrome operator members.
That's the summary. I'll post details of GAAC and RAeC membership privately to you, to save space here. If anyone else wants them, let me know.