The short answer: the two books recorded Dangun's mother differently
Let me give the short answer first. The Samguk Yusa and the Jewang Ungi recorded Dangun's mother differently. The Samguk Yusa recorded a woman who had been transformed from a bear. The Jewang Ungi recorded a granddaughter of Hwanung. The Gojoseon passage in Book One of the Samguk Yusa relates it this way: "The bear who kept the taboo for samchil-il (三七日, 21 days) turned into a woman, but the tiger could not keep the taboo and did not become human."
The Jewang Ungi relates a different picture. The comparative entry in the Encyclopedia of Korean Culture renders the passage this way: "That is, the Bear Woman of the Samguk Yusa disappears and in her place a granddaughter of Hwanung appears. Hwanung has his granddaughter take medicine so that she takes on a human body, and then has her marry Dansusin (檀樹神)." Another entry in the same encyclopedia records that he "had his granddaughter drink medicine so that she became human, and she married Dansusin (檀樹神) and gave birth to Dangun."
So why do the two records differ? According to the materials they relied on, the Samguk Yusa cited the Wei shu (魏書) and the Gogi (古記), while the Jewang Ungi cited the Bongi. However, no explanation identifying that difference in citation as the cause of the diverging accounts of the mother has been confirmed. The Encyclopedia of Korean Culture only summarizes the two Goryeo-period records as "similar in their basic content but showing differences in their detailed parts."
The Giipyeon of the Samguk Yusa: the earliest record, which cited the Wei shu and the Gogi
The Encyclopedia of Korean Culture points to the Giipyeon (紀異篇) of the Samguk Yusa as the earliest record concerning Gojoseon and Dangun. This record cited China's Wei shu and Korea's Gogi. The same encyclopedia records that no such content is written in the Samguk Sagi, the official history. So anyone who wants to check the Dangun story has to open the Samguk Yusa first. This can be seen as a condition worth knowing before comparing the two books.
Book One of the Samguk Yusa records the story of the bear and the tiger in relatively full detail. First, one bear and one tiger lived in the same cave. The two animals constantly prayed to Hwanung to make them human. Next, Hwanung gave them one bundle of divine mugwort and twenty cloves of garlic. Hwanung said, "If you eat this and do not see sunlight for a hundred days, you will become human." The ending, as seen above, was decided at samchil-il, that is on the 21st day.
The Samguk Yusa also recorded Hwanung's role. Hwanung commanded Pungbaek (風伯), Usa (雨師) and Unsa (雲師). He presided over grain and life, illness, punishment, and good and evil. He also presided over some 360 affairs of humankind, ruling and civilizing the human world. This passage can be read directly on the Samguk Yusa translation page at Wikisource. The account on the Jewang Ungi side can be checked in the corresponding entry of the Encyclopedia of Korean Culture.
Dangun's age of 1,908 in the Samguk Yusa and the 1,038 years in the Jewang Ungi
The two books also recorded the numbers differently. The Samguk Yusa records that Dangun ruled the country for 1,500 years. It then records that Dangun hid himself at Asadal and became a mountain god (山神). His age at that time was 1,908. The Jewang Ungi records that Dangun ruled for 1,038 years. And it records that he entered Mount Asadal (阿斯達山) and became a god, which is why he did not die.
The genealogies do not line up either. The Encyclopedia of Korean Culture compares the two records this way: "Although she is the same person in the sense of being Dangun's mother, on one side she is a woman incarnated from a bear, and on the other a woman incarnated from a god." The same entry also notes the number of generations. In the Samguk Yusa, Dangun is the third generation in the paternal line following Hwanin and Hwanung. In the Jewang Ungi, Dangun becomes the son of a woman belonging to the fourth generation of the family line running from Hwanin to Hwanung.
A different genealogy is also carried within the Samguk Yusa itself. The Goguryeo section of the Samguk Yusa cited the Dangun-gi (壇君記). That record relates that Dangun united with the daughter of Habaek of Seoha (西河) and had a child named Buru. The Encyclopedia of Korean Culture also notes the common substratum of the two records. The same encyclopedia summarizes that substratum as "a being that was not human takes medicine, obtains a human body, and becomes a woman."
The Jewang Ungi, published in 1287 and composed during the seclusion that began in 1280
Let us look at when the Jewang Ungi appeared. The Encyclopedia of Korean Culture records that the Jewang Ungi was issued only a little more than ten years after the Samguk Yusa. The same encyclopedia states that the Jewang Ungi was published in 1287. The book was reprinted in 1360 and again in 1413. The gap between the publication of the two books is not long.
The circumstances of its composition are also recorded. In 1280, Yi Seung-hyu submitted a memorial listing more than ten abuses, criticizing King Chungnyeol's misrule and the pro-Yuan power holders. That act earned him disfavor, and he was dismissed from office and went into seclusion. The Encyclopedia of Korean Culture records that the Jewang Ungi was composed during that period of seclusion.
The book's structure is also described. The second volume deals with the history of Korea. It is divided into two parts, the Dongguk gunwang gaeguk yeondae (東國君王開國年代) and the Bonjo gunwang segye yeondae (本朝君王世系年代). The same encyclopedia explains the motive for writing the Jewang Ungi as a sense of cultural crisis and a spirit of resistance arising from the overwhelming external pressure of the Mongols. That motive is recorded as being the same as the motive with which Iryeon wrote the Samguk Yusa in the same period. The encyclopedia also records that the Jewang Ungi incorporated the Dangun myth, transmitted through folk beliefs and ancient records, into the framework of Korean history.
The differences between the two records, set side by side point by point
Let us set the differences between the two records side by side, point by point. The first is the way Dangun is written: the Samguk Yusa used the character dan (壇) meaning altar, while the Jewang Ungi used the character dan (檀) meaning birch tree. The second is the documents cited: the Samguk Yusa cited the Wei shu and the Gogi, while the Jewang Ungi cited the Bongi.
The third is the mother. On one side she is a woman incarnated from a bear, and on the other a woman incarnated from a god. The fourth is the generation. The Dangun of the Samguk Yusa is the third generation in the paternal line. The Dangun of the Jewang Ungi is the son of a woman of the fourth generation of the family line.
The fifth is the division of Joseon. In its Gojoseon section, the Samguk Yusa recorded Dangun Joseon and Gija Joseon together. The Jewang Ungi described the Joseon of Dangun under the heading Jeon Joseon (前朝鮮, Former Joseon). Under the heading Hu Joseon (後朝鮮, Later Joseon) it mentioned the Joseon of Gija. And, including the Wiman Joseon that followed, it treated them as the Samjoseon (三朝鮮), the Three Joseons.
Summing up
To sum up. The Samguk Yusa recorded a woman transformed from a bear as Dangun's mother. The Jewang Ungi recorded a granddaughter of Hwanung, who became human after taking medicine, as Dangun's mother. The Encyclopedia of Korean Culture summarizes the two Goryeo-period records as similar in their basic content but different in their details. The two books also cited different materials. However, no explanation identifying that difference in citation as what split the accounts of the mother has been confirmed.
It is worth looking at the later records together with these. Among the records concerning Dangun in the Geography section of the Annals of King Sejong, the fragment of the story in question follows the Jewang Ungi. Moreover, the historical consciousness of a Dangun era, which began with the Jewang Ungi, was passed on to the reform-minded new literati of the late Goryeo period. When they founded Joseon, Dangun was enshrined with state ritual as the national progenitor. Official histories including the Dongguk tonggam also made clear that Dangun was the national progenitor and came to record him at the very head of Korean history. Reading the two records together, one can see this current emerge.
