Taimitsu is Esoteric Buddhism (Buddhism handed down secretly from general people) handed down by the Tendai Sect. It is named against Tomitsu (eastern esotericism), which is Shingon Esoteric Buddhism that has To-ji Temple in Kyoto as konpon dojo-seminary (basic hall of training), and was started by Saicho, who was the founder of the Nihon Tendai Sect (Japanese Tendai Sect).
Tendai Sect is recognized as exoteric Buddhism (teachings by Shaka which are made easy to understand for general people) because it has Hoke-kyo Sutra (Lotus Sutra) as its basis, or Konpon kyoten (primal scripture), but in the middle of the 9th century when Ennin and Enchin went to Tang and learned Esoteric Buddhism, the Tendai Sect in Japan also began to have the color of Esoteric Buddhism, too, and began to perform incantations of Esoteric Buddhism, and was started to be called Taimitsu. This is considered as the general opinion.
But the truth is, Taimitsu was founded by Saicho. Saicho went to Tang and learned mainly Tendai Doctrinal Studies, but it was popular to learn nenbutsu (Buddhist invocation), Zen, Buddhist precepts, and Esoteric Buddhism in China back then. Saicho brought back those teachings from Jungyo, Esshu, and tried to introduce them into Tendai Doctrinal Studies. Saicho was trying to establish original doctrinal studies as Japanese Tendai Sect, which has different meanings from Chinese Tendai Sect, but was based on the Chinese Tendai Doctrinal Studies founded by Chigi. Because of that, the Enryaku-ji Temple on Mt. Hiei is called the training hall of four sect studies.
Saicho gained the confidence of the Imperial Court, but when Kukai came back the next year with all the teachings of Shingon Esoteric Buddhism, he gained the confidence of the Imperial Court instantly, and played an active role, and Saicho realized that what he had learned was only a part of Esoteric Buddhism, and made great courtesy to Kukai, and became Kukai's disciple at Jingo-ji Temple on Mt. Takao with other disciples of his own. After learning sutra again and again from Kukai as a disciple, Saicho tried to use 'Rishishakukyo' (explanation of Rishu-kyo (Principle of Wisdom Sutra)) but was finally rejected by Kukai, and also, Saicho's best disciple Taihan and other disciples went over to Kukai and did not return back to Mt. Hiei, and for those and other reasons, exchange between Saicho and Kukai disappeared. Because of that, the aim of Saicho to establish Japanese original doctrinal studies of the Tendai Sect, which is said to be a synthesized Buddhism, with Buddhist precepts, Zen, nenbutsu, and Esoteric Buddhism all combined into Hoke-kyo Sutra, had to be interrupted there.
Further in the present day, the sects that sees Nichiren as the principle Buddha of Mappo (Age of the Final Dharma), determine Chigi and Saicho as marked bosatsu (Bodhisatta) in Zobo Period (Age of the Semblance Dharma) following the general opinion, while they criticize Ennin and Enchin in later history as they made hobo (criticism against one's Buddhism) by introducing Shingon Esoteric Buddhism into Tendai. But it is clear from historical facts that Saicho himself tried to introduce Esoteric Buddhism into Tendai Doctrinal Studies. Also, Ennin and Enchin just tried to complete the doctrinal studies of Tendai by re-studying Esoteric Buddhism and introducing it into Tendai Doctrinal Studies, because it was Saicho's desperate desire. Also, Enchin points out that there are five faults in Kukai's 'Jujushin Ron' (one of Kukai's masterpiece), and argued that 'neither Tendai nor Shingon is better than the other,' so the opinion of the sects that defines Nichiren as the principle Buddha have been completely denied here, too.
Later, it split into the Sanmon School, which followed Ennin, and the Jimon School, which followed Enchin. In Tomitsu, they teach sermons by Kongokai (Diamond Realm) and Taizo (Womb Realm), but in Taimitsu, they teach three great sermons, Taizo, Kongokai, and Soshitsuji (Wonderful Accomplishment).