I may be wrong, but I believe there are some sites you can only access with a membership to one organization or the other -- but whether any of that depends on whose CARD you have, I don't know. I sort of doubt it.
I have both manuals, but not where I am right now. There may be some slight differences in how protocols are taught. Once you are certified, you are going to have to go over such things with your buddies, anyway, because there are differences in how people do things.
The biggest thing with cave training is that you want someone who will truly TRAIN you -- the cave is an extremely unforgiving place, and you want your instructor to hold you to "cave" standards, so that you understand what they are. I have taken classes from four different people from three different agencies, and not all instruction is the same.