Mine is the ATA.
I started in the WTA (GM Sun Duk Song's organization), and ended up leaving to an ATA school purely due to the instructor, nothing to do with the orgnization. Once in the ATA, I found my new fellow students were just learning the new Songahm forms, having just switched from the forms I was doing before. (low block, punch, low block punch, low block punch.. sound familiar? hehe).
The ATA is set up on a few levels, of course you have the school level at which most students stay pretty comfortably at. At the school level they learn a standard curriculum (obviously with some varience as instructors add things), with set forms, testing requirements, equipment etc.
From the school owners' perspective though, things are much different in terms of most of the other organizations that I'm familiar with. The Headquarters is very accessible and available to the owners, you can call any number of Masters and get resolution if it the situation isn't clearly illustrated elsewhere. From what I've noticed, the 'network' is just much more prevalent than most other organizations. It has a clearly shown chain of command, with rank leading upwards etc. While there is some 'McTKD' aspects, with things being packaged etc, I think it's unavoidable with the amount of standardization the organization requires. I'll see if I can't illustrate those more clearly in a bit.
From a regional level you have regional officers, a 'buck stops here' person for each region (US has its, Canada/UK have the WTTF, SA has STF, all with the same set up, much like the U.S with states, gov body). The Regional VP is the 'go to' in the area, who helps coordinate, assists schools, basically the overseeing person of any given region. The RCT (Chief of Tournaments) of each region then coordinates those (huge job with the divisions, points etc).
In any given region you will have a number of schools that are all mostly autonomous, but they do work together at times with regards to regional events, sometimes marketing coordination, testings, etc.
On the national level you have the headquarters (I suppose international level as they also oversee the worldwide Songahm organizations). There, they oversee instruction, forms, one-steps, requirements etc. They have departments such as tournament, testing, instruction, protech (weapons training), etc. Each with a presiding Master/chain of command. At the top we have the Masters Council, a group of the 'heads of state' so to speak. These masters help shape the changes in the organization and determine it's direction.
So back to what it gives me personally? (I know, a lot to digest and I only briefly touched on the aspects).
I get a set curriculum for advancement, a clear path that I can take at whichever speed I deem is mine. I get a standard as far as how techniques are performed, variations of course, but a set of guidelines to measure. I get a network of people that are closely linked, each school owner _knows_ the other school owners in the region, if not the surrounding regions as well. We all work together for events, testings etc (I sit on usually 12-18 testing panels per year around the state). I get a sense of family?
I guess it's somewhat difficult to clearly illustrate all of the points or aspects, and I'm sure I've overlooked other organizations similarities as I haven't been in a different one for some 22 years. I do other styles as well for my own growth, but none of those in any org.
A quick summation of the history would be: ITF with GM Choi brings TKD to the US (one of the people, I know GM Rhee and other groups). The ATA is formed as the NGB of the ITF in the US. Groups splinter as 'too many chefs in the kitchen' (my own grasp of it of course).
As far as groups that started in the ATA, I know of the ITA, USTA, HTF, and quite a few others. When a group of people have different ideas and split off to form their own organization. I have many friends in the other organizations and I'm sure I'll make many more before I'm gone.
I've rambled quite a bit, but I'd be happy to more clearly illustrate points or aspects if people find parts lacking (heck, I know I do heh).