From the Agency for Cultural Affairs and the Japan Arts Council.
Noh is a drama of song and dance, perfected around the 14th century. The story moves forward through dance-like movements, with chanting (utai) and instruments (hayashi). Its very simple, set style shows human feelings with great delicacy.
Kyogen is a comedy told in words. It shows the many kinds of laughter found in the lives of ordinary people.
Nogaku later had a great influence on bunraku puppet theater, kabuki, and even today’s arts. It was named an Important Intangible Cultural Property in 1957, and added to the UNESCO list in 2008.
The roots of Noh and Kyogen lie in sangaku, popular entertainment brought from China in the Nara period (8th century). It mixed acrobatics, magic, song and dance, music and mimicry.
Mixed with the older arts of Japan, it came to include funny mimicry and short plays, and its name changed to sarugaku. From the Heian to the Kamakura period, its song and dance were carried on at temples and shrines.
Among the troupes of sarugaku, four troupes based in Yamato (today’s Nara Prefecture) were especially active. They seem to have led to today’s schools of Noh.
Kan’ami (1333–1384), who led a sarugaku troupe in the Muromachi period, took in parts of dengaku and other popular arts of his time. With his son Zeami, he perfected Noh.
The third Muromachi shogun, Ashikaga Yoshimitsu, watched Kan’ami and the 12-year-old Zeami perform, liked them very much, and protected them. Zeami made Noh a drama centered on beautiful song and dance, and wrote many works on the art, such as Fushikaden.
Kyogen, which was growing into a comedy of spoken lines, joined the Noh troupes, and came to be performed in turn with Noh.
There is no single right answer. Talking it over with someone at home is a good way to learn.