In Korea, there were several martial arts groups after independence from theJapanese. All wanted to get recognized.  General Choi being in the military had the dominance and called it TKD, and tried to unify all factions into his style called Oh Do Kwan and ITF. He failed and another rival organization WTF led by
Dr. Un Young Kim, with backing from the civilian government succeeded in unifying all the TKD factions. General Choi courts South Korean government to support his organization, but is turned down. He then courts North Korean government, gets support but also leads to the weakening of  his ITF organization by massive departure of his subordinates.  WTF takes this opportunity and takes the void left by ITF and leads as the  official governing body of TKD . 

The WTF emphasis is on tournaments, ITF on traditional TKD in sense of the martial art . NEITHER IS BETTER THAN THE OTHER. Most students are not really interested in the politics unless blindly swayed by their instructors or are not exposed to both styles.  The future trend for TKD appears to be growing into a versatile martial art, which Gen. Choi achieved briefly in the military in the  70s before its decline. But whether WTF/ITF or other TKD organization, it is slowly getting there. The trend speaks for itself; just about all the WTF Korean gold medalists in the Olympics had multiple BBs in other martial arts; and year after year, more and more weapons and hapkido techniques are demonstrated by the WTF demonstration teams.

The Olympic acceptance

In Korea, the remaining  TKD factions that lost out to General Choi
underwent a significant reform. WTF was formed about the same time ITF was formed. Impeded by internal politics and lack of international experience, pre-WTF organizations tried to create a network like ITF with little progress. But WTF did carry out a massive reform of  TKD in Korea to distinguish itself from ITF and also as an anti-General Choi style campaign. It created the Pal Geh forms; the ITF styles and movements were phased out in the military. The military modified the ITF TKD moves with a speedier movements and more incorporation of
WTF movements and philosophy.  The TKD terms were changed from ITF s terms such as  soodoh (knife-hand) of Japanese origin to  sohn-kahl ,  hah dahn mahki (low block) into   ah rae mahki ,  dae ryun (sparring) into  gyorugi  into pure Korean words. Some other terms are also changed.  The patterns are now called poomse  instead of   hyung . The WTF and ITF separately try to enter the Olympics but is stalemated.
Strengthened by the weakening ITF, WTF did a massive campaign to capture the void left by lack of leadership in world TKD arena.  Korean government appoints a new man for the task, Dr. Un-Yong Kim.
Dr. Kim, an academic scholar with several graduate degrees from both in Korea and the USA was a man of action for the job. A speaker  of 6 languages, a seasoned former diplomat, excellent people skills, and with strong political ties both domestic and abroad, was a doer.   As a director of 100% government backed WTF, he united some 36+ TKD factions(big and small) in Korea under one banner by organizing a non-biased governing body.  A feat Gen. Choi tried to accomplish over two decades was done in matter of few years. Most important of
all, he built the WTF(Kukkiwon) headquarters building within a year to edify WTF by securing support from private sectors through sheer diplomacy. His work encountered initial resistance and alienated some heads of different TKD stylists,  but his unpretentious loyalty to the heads of different factions and connections facilitated the creation of a civilian backed WTF.

The acceptance of WTF TKD into the Olympics as an demonstration sport in 1988 was also through the  diplomacy of  Dr. Kim. The demonstration itself  became a catalyst for the legitimate acceptance and growth of  WTF TKD by the world. The  real  TKD sanctioned by WTF spreads via the television broadcast of Olympics
to the world media is etched into the world viewers as WTF = real TKD. From this point on, WTF begins to grow and dominate the world TKD arena. It also announces that it will take in other styles of non-ITF TKD stylists. Always quelled and snubbed off by ITF, they joined the ranks of WTF organization. Some who did not agree with WTF politics later reverted back to their own styles, but this time, did not have to be constantly aware of  ITF TKD s snubbing attitudes.

General Choi upon hearing the news feels  taken advantage of   what was justifiably his work with his opponents taking credit for his work. It becomes a severe blow for him, for what he had spent so much of his life on, was gone.  One of his lifetime dreams, the inclusion of  ITF style TKD into the Olympics has now become an illusion.

ITF assessment of  WTF stylists:

All they do is fancy foot and legwork with a lot of sound. No real power. Everything is geared to competition and tournaments. No spiritual arts, and a lot of good foot techniques such as twisting front kick(a kick that springs up from a standing position both feet facing forward, foot is twisted inwards and kicked up towards the opponents nosebridge at an angle)are lost, and everything is competition oriented. The practicality of the martial arts is lost for speed and points in tournaments. Upper body and hand movements are not emphasized nor does it call for hitting/striking the face/head as in real fighting, or close range when someone is tackled.  Thus the students are only skilled in playing  tag points . No real practical value in actual fighting. The head of WTF is not a TKD practitioner but a bureaucratic figurehead which is why WTF certificates are signed Dr. Un Young Kim. He has neither demonstrated nor taught TKD.   The WTF patterns are of no aesthetic value, it is just a breakdown mix and match of ITF forms. No beauty in forms. Taekwondo?  Gaekwondo  is more like it.(Note:   Gae  is dog in Korean, loosely translated, it means  dog pawing ).

WTF assessment of ITF stylists:

Outdated and limited kicking techniques taught by old masters that have no practical application & imagination for WTF sparring. Non contact or light contact  pussy footers -because one is conditioned to fight non-contact or light contact,  they get blown away in  real  hard full contact sparring situations. Impractical sparring moves-eg. blocking instead of dodging. Moves are too slow for practicality and vulnerable. (e.g. plain wheel kick, where the heel is turned in an arc towards the opponent s head by turning back without hooking at the knee ).  Too low and wide stances that slows down each movement and telegraphs your move. No power nor speed in moves. No efficient aerial kick techniques that require high skill which can be devastating,. No innovations or development in their moves. Thus the students have limited skills.  ITF TKD? Gimme a break.

Rules of WTF(simplified terms).

WTF sparring is tournament oriented.

1. Hits must be on the certain target areas above the waist line for points.
2. No punches allowed to the face. If punch is done to the body, one must hit to the body with a  forcible impact to the opponent.
3. Kicks to the head allowed.
4. No grabbing or holding the opponent.
5. No points for hitting in the back.
6. No sweeps or take downs. 
7. Protective gear (including forearm, shin, head, chest, and groin protection )and now doboks must be sanctioned by WTF

 Rules of ITF(simple terms).

ITF is traditional training oriented. Tournament Rules
(Again, this is general and varies in each splintered ITF-style group).

1. Protective gear required(the level of protection depends on the instructor).
2. Light padding on shins, knees, foot, and arms.
3. Light head gear.
4. Timed sparring.
5. Points for certain targets including hand techniques. Stopped for points(in some).
6. Non, light, or medium contact. (Full contact is rare, but is allowed in some tournaments). 
7. Chest protector(depends on instructor).

The real difference between ITF and WTF
ITF being traditional adheres to an  equal emphasis  on patterns, breaking, sparring, & self-defense techniques. WTF has  more emphasis  on sparring aspect than patterns and self-defense. That is not to say WTF does not have patterns or self-defense. Breaking in WTF however, is not much emphasized as breaking in ITF.

Which is better?  In my opinion, the objectives of ITF and WTF are different. ITF(traditional martial art TKD) and WTF(tournament geared TKD) are two different systems that has diverged from the same origin.  The best answer would be the better TKD stylist. The question is more like who is the better player? Comparing the better squash player vs. the better racquet ball player IMHO does not do justify who is better, even though the  game may be similar.