If the airport you're based at has federal grant obligations (basically if they've taken money from the Airport Improvement Program) then being a "flying club" by the FAA's definition entitles you (as a group) to the same rights as any other operator at the field (like the right to fuel your aircraft with fuel you buy yourself & have trucked in, or the right to do maintenance on the aircraft), subject to the same rules and regulations the airport lays down for all operators.
The biggest material difference is that "flying clubs" aren't considered to be "commercial" ventures by the FAA (and by extension grant-obligted airports).
As an example, let's say you have a flying club & you "charge" $50/hr for the use of one of the planes, and your airport management decides that since you're charging money for the use of the planes you're a commercial venture, and need to have commercial insurance and the like.
If you meet the FAA definition of a "Flying Club" and the airport has federal grant obligations you can ask your local FSDO to step in and effectively tell the airport management to get stuffed - as a flying club you're treated more like an individual than a commercial business, and the airport is not allowed to impose those commercial requirements on you.
If you don't meet that definition the FAA won't intervene on your behalf (as far as they're concerned you are a rental outfit / commercial FBO at that point and the airport is correct in treating you as one).
Note that you can still call yourself a "flying club" even if you don't meet the FAA's definition -- I belong to a flying club that has no aircraft (we're just a bunch of folks who like to talk about flying and organize the occasional fly-out event or safety seminar - some of us own planes, and some rent from local FBOs), and it doesn't really matter to us that we don't meet the FAA definition of a flying club because we don't have any interaction with the airport where that designation would benefit us.
If you're the kind of organization that has aircraft available for the use of your members however it probably makes sense to ensure that you're organized & run in such a way as to meet the FAA definition of a flying club.