First, this will vary between styles and even school to school, or organization to organization.
But, to do a quick summary using Shotokan as a template.
1) Kata: The katas were revamped when they went over to Japan to focus more on the "physical education" benefits of karate and many things were altered, so some things are not shown as clearly as their Okinawan counterparts. For example, Wansu kata was known for a "fireman's carry" style throw in it. This is now a jumping/spinning move in Empi kata to enhance athleticism.
2) Kihon: In Okinawa, most of those moves were embryonic and had multiple layers. They could be used as blocks/strikes/grappling techniques. When it went over to Japan and also as karate spread to military men, the emphasis really became the lowest level of application and things were not always fully explained. For example, chambering to the waist was/is now explained that it is used to get more power instead of the actual application that the withdrawing hand was holding something from the attacker, like a limb or clothing and you were pulling him into the strike. A block was not always a block and could be used as a joint lock or part of a stand up grappling application.
3) Kumite: Originally, karate was a civilian self-defense system. It was mean to overcome the most common types of attacks, it was NOT designed to be used on a battlefield or in a sports context. This meant that the kata and techniques were close quarters and not a longer distance. As popularity grew and the want for sparring increased, Shotokan patterned their kumite after the sparring found in kendo. This is why you have a larger starting distance than you would in the street. It also meant that techniques tended to be more of a longer range. Due to this change, the techniques found in kata don't make sense because the attack and distancing is very different.
Many of these changes were also brought back to Okinawa as karate started to be taught more publicly and after WW2, the Okinawans could make a very good living by teaching US serviceman and adopted the changes as well.
This is not as clear cut as the summary makes it, but it can give understanding as to why something regarded as so effective seems to have such a disconnect with how we view fighting.