The name "Taekwondo" was given to the art by the man who is largely regarded as the "Father of Taekwondo" General Choi Hong Hi in 1955. In 1966 Gen.Choi created the ITF in South Korea. The newly formed ITF had connections with several nations who were all considered the founding national memebers. The main function of the ITF is to coordinate and approve tournaments and seminars, set standards for teaching(patterns,sparring,etc.), and to service its members in regards to ranks and certifications. The ITF style is based more in the traditions of the art as it came to be in Korea. In a dojang(school) that teaches the ITF style you will see more emphasis on technique and control. The process of learning is the big focus with the result being a byproduct of good training and not the main focus. This however does differ from school to school to varying degrees as each master may take a different approach.
Forms are reffered to as tuls(patterns). Each pattern is named after a notable point or person in korean history, and the movements and techniques in the patterns reflect this. In sparring the student is expected to show a high level of the control and technique that has been learned. In competition points can be deducted for blows that debilitate a foe which is an emphasis on controlling ones self. It takes a high degree of strength and skill to commit to a full force attack and then pull that force and check yourself at the moment of impact. Punches to the head and face are allowed in competition which is always semi-contact(emphasis on control).
The WTF was established in 1973 when General Choi left Korea for political reasons and took the ITF headquarters to Canada. WTF is what is featured in the olympic games as the face of Taekwondo. The WTF only governs the function of Taekwondo competitions and as such is more of a competition committee than an actual style. However there are some fundamental differences between WTF and ITF. The first of these is that forms are reffered to as Poomsae. WTF teaches less patterns to the student while placing a big emphasis on competition. The focus of training is more on the result and less about the process of acheiving the result. Sparring is full contact. Points are awarded for knocking out an opponent, so there is less emphasis on control and more on full force strikes on the opponent in an attempt to disable them. Punches to the head and face are not allowed. For theses reasons WTF is sometimes reffered to as a martial sport rather than a martial art.