Iaijutsu is the art of killing on the draw. Iaijutsu teaches how to draw quickly and in such a fashion as to negate an opponents attack with finality.
Seitei-gata iaido (that set of techniques recommended by the ZNKR) is like a moving meditation - the draw and cut are very deliberate, formalized and beautiful. It is as far removed from iaijutsu as kendo is from kenjutsu. Iaijutsu is more direct and forceful, less concerned with the state of the practitioner's mind and more with dispatching the opponent.
Having said that, iaido schools are generally affiliated with a particular ryu of iaido. In addition to the seitei-gata, students also learn their own ryu's techniques, which may be close to the seitei-gata in feeling or close to what is described here as iaijutsu. It's not completely black and white.
In the latter half of the 15th century, Ienao Izasa (also known as Choisai Izasa) founded the Tenshin Shoden Katori Shinto Ryu. Together with his leading swordsman, he devised the art of attacking with the draw called iai-jutsu. In the early part of the 16th century, the Tatsumi Ryu and Takenouchi Ryu also practised iai-jutsu.
In the late 16th century, Shigenobu Jinsuke allegedly was divinely inspired to develop a new sword-drawing art. He renamed himself Hayashizaki after the inspirational place and founded the Shimmei Muso Ryu to teach his art, called batto-jutsu. He was one of the first to teach swordsmanship as a way for spiritual development. Popularly misidentified as the originator of iai-jutsu, his influence has been great. More than 200 ryu have been founded in the wake of Jinsuke's inspiration and image, many of them named after him.
Various headmasters in the line of Jinsuke's teachings formed their own ryu. Among them were Shigemasa Tamiya (Tamiya Ryu), Kinrose Nagano (Muraku Ryu) and Eishin Hasegawa (Muso Jikiden Eishin Ryu), who were the 1st, 3rd and 7th headmasters descending from Jinsuke. The ryu which branched out from the teachings of these and others are too numerous to mention here.
Hakudo Nakayama, who lived at the beginning of the 20th century, studied Omori Ryu, Muraku Ryu and Muso Jikiden Eishin Ryu and was experienced in all aspects of swordsmanship. He became the 16th and last undisputed successor to the Jinsuke/Eishin line. He also studied Shindo Munen Ryu and Yamaguchi Itto Ryu. He went on to develop his own style, Muso Shinden Ryu batto-jutsu. Due to his diverse experience, the ryu boasted a bewildering array of techniques. He was asked to develop a simplified curriculum. He did so, and made the techniques available to all interested persons, largely kendoka. These forms of iai-jutsu, along with others, were gradually restyled as iaido in the late 40s.
In 1967, the Zen-Nippon Kendo Renmei formed a committee to develop a standardized curriculum of study for iaido. This curriculum was to be recommended as study to students of kendo, who were losing touch with the dynamics of combat with real swords. Members of Muso Jikiden Eishin Ryu, Muso Shinden Ryu and Hoki Ryu recommended a curriculum of seven kata that became known as the seitei-gata. In 1977, another committee from the same ryu plus Tamiya Ryu added three more kata to the seitei-gata. The seitei-gata iaido has the largest popular following in Japan and abroad. The Zen-Nippon Iaido Renmei was formed in 1948, and has done a great deal of work to promote iaijutsu and iaido. It has its own autonomy and standards.
Only a handful of ryu are represented by the major organizations; thus the hundreds of traditional iai-jutsu ryu did not contribute to the foundation of iaido. Classical iai-jutsu exists today but largely goes its separate way from iaido.