This point has probably been made in other posts, but it is worth repeating anyway.
The Kukkiwon is a Korean organization that establishes the technical standards for the MARTIAL ART of Taekwondo. The Kukkiwon is concerned with promotion standards, dan/instructor certifications, and TKD curriculum via the Kukkiwon textbook. The style of TKD promoted by the Kukkiwon is often called "Kukki Taekwondo".
The WTF is an international sports federation that exists to govern the Olympic sport of Taekwondo gyoroogi (sparring). Most of its activities revolve around certifying officials (i.e. referees), establishing rules, holding and promoting events for said sport.
The style of TKD promoted by Kukkiwon is often called "Kukki Taekwondo". So then, the Kukkiwon is a MARTIAL ARTS organization promoting the MARTIAL ART of Kukki Taekwondo, while the WTF is a SPORTS organization that promotes the SPORT of Kukki Taekwondo sparring.
The WTF does not give out Dan certificates, invent forms, or provide testing standards. The Kukkiwon does not make up rules for Olympic sparring, certify refereees, or decide where the next World Championships will be held.
If there is such a thing as "WTF Taekwondo", then the most accurate definition would be that "WTF Taekwondo" is the sport of TKD sparring as governed by WTF rules. Therefore, WTF TKD is that sport that is concerned primarily with kicking where you wear all those pads and "never use your hands." ;-)
On the other hand, Kukki Taekwondo is the martial art comprised of kicking/punching, forms, breaking, sparring, and self-defense principles. Just look at the Tae Geuk and Pal Gwe forms and you'll easily see they are about 90% hand techniques (some of the Dan level forms have NO kicking), including punches, uppercuts, backfists, fingertip strikes, hammer fists, throat strikes, and elbows. Go to a reputable Kukki TKD school, and they will require mastery of hand and foot techniques, breaking, sparring, one-steps, and philosophy/history.
The confusions between Kukki TKD and "WTF TKD" or "Olympic TKD" is that the Kukkiwon and WTF occupy the same building, and at one time the same person was the president of both. Additionally, in America, you have schools that say they practice "WTF TKD" or "Olympic TKD", when they really mean "Kukki TKD". Of course, there are also those schools who say they practice Olympic/WTF TKD, and they are actually correct because they cater exclusively to competition and athletes (i.e. a lot of schools that call themselves "sport TKD"), and forego the full Kukkiwon curriculum.
I second von1's comments regarding self-defense. Sport/WTF/Olympic TKD by definition is a sport, and thus the very limitations imposed on it for competition would render it less effective for SD than a full-blown martial art. On the other hand, Kukki TKD provides many techniques which would be applicable to SD situations. A TKD practitioner who is versed in the full spectrum of techniques available in Kukki TKD would have many weapons available to them, and it's absurd to assume that such an individual would just bounce around and try to take out a mugger with jump spinning hook kicks to the head.
_Taekwondo: State of the Art_ by Masters Sung Chul Whang, Jun Chul Whang, and Dae Sung Lee have a good treatment on the topic of the relevency of sparring to self-defense. Without reprinting their entire argument, a well-trained TKD athlete will have developed very fast movements, powerful strikes, effective footwork, a good sense of distance/timing, and will know how to take a hit and keep fighting. Now, give that same athlete more targets than just the head and body (knees, throat, eyes, groin, etc), and more weapons (punches, elbows, knees, etc), and I'd say that the athlete will probably do a lot more damage than someone that just sits around doing forms or one-steps all day.