The Tale of the Bamboo Cutter, also known as the Legend of Princess Kaguya, is by far one of the most influential and famous Japanese folk tales to ever exist. In fact, it is considered to be the oldest Japanese narrative in existence.
Basically, the story is about an old man (the titular Bamboo Cutter) who, after cutting open a mysterious, glimmering stalk of bamboo, discovers a tiny, beautiful infant hiding within. As the bamboo cutter, named Taketori no Okina, and his wife are childless, he happily takes the tiny girl home to raise and care for with his wife; they name her Kaguya-hime, or Princess Kaguya. Under the loving care of her foster parents, Kaguya grows up to be a full-sized woman whose radiant beauty is out of this world. Taketori tried to protect his daughter from outsiders, but so great was her beauty that news of her eventually spread far beyond her home.
Eventually, five princes came to Taketori’s home and asked for Kaguya’s hand in marriage. After convincing Taketori to let Kaguya choose among them, Kaguya, not wanting to get married to a total stranger, concocted five impossible tasks for each of her suitors; she told them that the suitor who managed to bring her his requested item would win her hand in marriage. Later that night, Taketori told the princes what they had to seek. The first was told to bring Kaguya the stone begging bowl of the Buddha from India, the second a jeweled branch from the island of Hōrai, the third the legendary robe of the fire-rat of China, the fourth a colored jewel from a dragon's neck, and the final prince the cowrie which was born from swallows.
None of the princes succeeded. The first three attempted to fool her with fakes, which she saw through, the fourth gave up after encountering a storm, and the fifth lost his life during his quest. Eventually, the Emperor of Japan, Mikado, visited Kaguya and fell in love with her. Although she didn’t make him preform any tasks, Kaguya rebuffed his proposals, explaining that she was not of his country and thus could not go to the palace with him. She stayed in contact with the Emperor, but continued to rebuff his requests and marriage proposals.
That summer, whenever Kaguya saw the full moon, her eyes filled with tears. Her adoptive parents, frantic with worry, tried to find out what was wrong, but she refused to tell them. Her behavior grew stranger and stranger until she finally broke down and confessed to her parents that she was not of this world; in reality, she was a celestial maiden from the Moon and soon had to return to her people. The people of the Moon were waiting for her and she knew that she would have to leave her home, her parents, and her new love soon.
The Emperor sent many guards to her house to protect her from the Moon people, but when an embassy of Heavenly Beings arrived at the door of Taketori no Okina’s house, the guards were blinded by a strange light. Kaguya announced that, although she loved her many friends and her family on Earth, she must return with the Moon people to her true home. She wrote sad notes of apology to her parents and to the Emperor, then gave her parents her own robe as a memento. She then took a small taste of the elixir of life, attached it to her letter to the Emperor, and gave it to a guard officer. As she handed it to him, the feather robe was placed on her shoulders, and all of her sadness and compassion for the people of the Earth were forgotten. The heavenly entourage took Kaguya back to Tsuki-no-Miyako (“the Capital of the Moon”), leaving her earthly foster parents in tears.
Kaguya’s parents, stricken with grief, became bedridden. The officer returned to the Emperor with the items Kaguya-hime had given him as her last mortal act, and reported what had happened. The Emperor read her letter and was overcome with sadness. He asked his servants, “Which mountain is the closest place to Heaven?”, to which one replied the Great Mountain of Suruga Province (known to us as Mount Fuji). The Emperor ordered his men to take the letter to the summit of the mountain and burn it, in the hope that his message would reach the distant princess. The men were also commanded to burn the elixir of immortality since the Emperor did not wish to live forever without being able to see her.
In Sailor Moon:
Princess Kaguya and her story are referenced quite a few times in the Sailor Moon series. The most obvious example is from the second movie and the manga side story The Lover of Princess Kaguya, with the debut of the main villain, Princess Snow Kaguya.
However, Snow Kaguya is not actually the Celestial Maiden from the folktale; she was just named that by the man who discovered her comet, Kakeru Ozora.
Later on, Luna, in her human form, assumes the name Princess Kaguya.
Also, Kakeru believed that the woman speaking to him in his dreams was Princess Kaguya, only to wake up and discover that it was his long-time girlfriend, Himeko Nayotake.
And finally, Human Luna refers to Himeko as Kakeru’s “true” Princess Kaguya, telling him to return to her and live a happy life together.
Alright, so that covers the explicit mentions of Kaguya. But what about the rest?
Princess Serenity herself, although primarily based on the Greek goddess Selene, can be seen as an analogy for Kaguya. Like Kaguya, she is a Celestial Maiden and Princess of the Moon (and, in some tellings, Kaguya is actually the Moon’s Princess) who goes to Earth and falls in love with a mortal ruler (the Emperor in the folktale, Prince Endymion in Sailor Moon).
Also, episode 40 contains a few more references to Kaguya. For one, Usagi cries when she sees the full Moon, just like Kaguya once cried for her lost home. However, Usagi’s not only crying out of sadness (for her lost love, Endymion/Mamoru), she also cries out of happiness at the Moon’s beauty.
Finally, the story of the Legendary Lovers is about a Celestial Maiden who came to Earth and found love with a mortal.
That’s about it, I think!