From a question on the Alternate History Online facebook forum:
Kublai Khan’s invasion fleet is wildly successful, and establishes a firm beachhead on Hakata Bay, from which they subjugate the island of Kyushu by 1275. From there, they conquer Shikuku and southern Honshu, working their way steadily northward, armed with bows and hand-held explosives against which the Japanese forces have no defense. The only thing that limits the Mongol expansion is the expense and danger of getting people and material from the Asian mainland to Japan. Those darn winds!
Peace comes in 1287, when the Retired Emperor Kameyama, seeking to cut support out from under his rival, the Retired Emperor Go-Fukakusa, persuades his son, the reigning Emperor Go-Uda, to formally acknowledge the Mongol Great Khan as Shogun, the supreme commander of the Japanese military government. The current military government at Kamakura is defeated, and flees northward, forming the Shogunate of Mutsu in northern Honshu and *Hokkaido (then called Yezo).
As a new vassal of Yuan dynasty China, Japan is governed by a sub-Shogun at Bakutou (its Japanese name, near OTL Fukuoka) on Hakata Bay. The Emperor in Kyoto is a mostly ceremonial title, although the office does have jurisdiction over the government of the city of Kyoto.
Emboldened by his success in Japan, Kublai Khan orders more invasion fleets built and deployed, conquering Hokkaido (and the shortlived Shogunate of Mutsu), Formosa, Luzon, and Pagan. In doing so, however, he beggars his own court. After the failure of the fourth invasion fleet to conquer Borneo and the humiliation of Admiral Omar in Dai Viet, Kublai’s generals defect to Nayan Khan, who defeats Kublai in single combat and assumes control of the Mongol Empire and Yuan China.
Military support for the puppet government in Bakutou dries up, and civil war in Japan breaks out in 1298. When the dust settles, there is a new military government ruling from Edo. The Hong Shogunate pledges allegiance to the Great Khan, but writes court documents in Chinese and speaks Korean. The grandson of the the first Hong Shogun, however, speaks Japanese.
Unlike the Tokugawa Shoguns, the Hong Shoguns never close off Japan to foreign influence. Even after the fall of the Yuan dynasty, they continue to trade enthusiastically with the new states in Korea and China, as well as with European explorers.
This opens the Hong Shoguns up to conversion by Portuguese missionaries, and in the mid 1600s, there’s a Christian rebellion that cripples the Shogunate. The Emperor, however, forms and alliance with the Buddhist forces of Mount Hiei, and crushes the Christians, establishing a new military government at Mount Hiei. (Potential point of divergence: if the Japanese Christians manage to get Spanish support, we could end up with Japan following in the footsteps of the Philippines).
The Hiei Shogunate closes off the country, and so it remains until American pressure brings about an imperial restoration in the 1880s. The imperial seat to Tokyo (which is actually called Tokyo, but isn’t in quite the same place) and proceeds to cultivate an interest in trains and overseas colonization.
TL;DR: We get a Japan more or less like the one in our time line, but with more Korean and Chinese influence. Buddhism played a bigger role, as did Christianity (before it was crushed). Modern Japanese history parallels ours, although the food served in Tokyo is a much spicier.