BFCs tend to be volunteers - or at least not to be paid by the women they are helping. I know that in the BfN that is a rule really, I suspect it is in the NCT/LLL/ABM. LCs may well do volunteer work but they can also charge for their time.
LCs are more problem-solving oriented, I think - BFCs tend towards providing, as others have said, information and support, not advice. (Although the mothers seem to hink they've had advice.)
The BfN, iirc essentially split off from the NCT because of an argument about funding/advertising. Their courses are (from what I know) free, with creche provided, but you are required to help out on the phone line.
I think even IBCLCs are tied to an orgnisation of sorts - as they should be. Anyone providing any sort of help should be tied to a professional body of some sort, so that complaints/issues/etc can be properly handled.
If you are ever helping mothers one-to-one, having supervision (which I believe all the four organisations named provide) is essential. If a situation alarms or upsets you, it's vital to have someone you can seek help from.