Already by the 5th c. BCE, philosophic and scientific ideas were taking hold among the Greek intellectuals, and with that, the power of the gods of Homer and Hesiod was greatly reduced. This did not mean that Homer or Hesiod were no longer read; there were professional performers whose repetoire were the Homeric poems, which they would recite from memory at festivals. Poets and dramatists still revisited the old stories, though in some cases (Euripides, especially) with increasing skepticism. When Alexander spread Greek culture throughout the Mediterranean, and established a new city of culture in Egypt (Alexandria), where there would be a great library and research center (the Museum ["Home of the Muses"]), the stories still had emotional power, even if few believed in the exact cosmogony set forth in the Theogony or Homer.
During the Hellenistic period (3rd c. BCE - 1st c. BCE), scholar poets often outdid one another in highly allusive verse, which showed just how educated they were. Often instead of saying things directly, poets would use roundabout ways of expression. One example mentioned in the book is where Zephyrus, the gentle west wind, is not referred to by name, but called the "twin brother of Ethiopian Memnon." Such poets were not concerned with moving the general public, but only other educated people like themselves.
In that rarefied atmosphere, there also came to be written handbooks of mythology, sometimes focusing on particular themes (e.g. metamorphoses, or constellations). One can see that Ovid, with his own Metamorphoses, came out of that tradition. As Greek scholars studied the heavens, myths came to be associated with various constellations in the night sky. Roman poets largely inherited this scholarly poetic tradition, even some like Vergil manage to make their poetry seem fresh.
Finally, Christianity won out over pagan worship in the Roman world. The mythological significance of the stories no longer held sway (at least on the conscious level). Considering the triumph of Christianity, it is amazing that ancient pagan literature has survived at all. Keep in mind the difficulty of maintaining manuscripts. There was no careful treatment of manuscripts; they wore out just as books do in our public libraries. Books that mattered would be copied in several editions (all by hand), and when editions wore out, copies would be made to replace the old volumes. For some Christians, pagan literature was to be avoided at all costs -- it represented a value system they didn't want to support. Others loved the literature and the often exquisite method of expression, so that, even as they were criticizing the pagan religion, pagan works would be copied. Some authors, like Vergil, or Cicero, were such masterful stylists, that the argument of literary style as something to be emulated was advanced. Still, it's amazing that as much of ancient literature survives as does.
When vernacular literature (in the everyday tongues of English, Italian, French, etc.), the authors looked to ancient literature for guidance and for storylines, so that some of the ancient stories got new lives. Of course, these stories would often undergo transformations that were often great. Stories about the Trojan War, for example, would have Achilles, Ajax, Hector and the rest, acting like medieval knights, following a code of chivalry. See, for instance, the quotation M & L have from Chaucer's "The Knight's Tale" on p. 552. Even if the stories underwent great changes, the manuscripts continued to be copied until the invention of the printing press assured the continued existence of what ancient literature survived the Middle Ages.
In the Renaissance, with the reintroduction of classical Greek literature in the West, and the clear admiration of poets like Petrarch and Dante for their ancient counterparts, there was a resurgence of interest in classical art and thought, and a rebirth of scholarly study of ancient languages and literatures. And that interest and study has continued up until the present time. Starting with the Renaissance, though, there is less attempt than in Late Antiquity or the Middle Ages, to use Greek and Roman authors as evidence against pagan worship and to glorify Christianity. Classical works, though still seen through the lens of the current period, came to be valued in themselves.
One of the first poetic works written in the Americas was George Sandys' translation of Ovid, Ovid's Metamorphosis [sic] Englished. Though they depended more on philosophy and history, the Founding Fathers looked to ancient Greece and Rome (especially Rome) for inspiration. Latin and Greek were required subjects for anyone going on to University, while moralistic and bowdlerized versions of the myths were read by many (e.g. Nathaniel Hawthorne's Tanglewood Tales).
The interest in ancient Greek literature was high during the Romantic period (late 18th c. and early 19th c.) for poets like Goethe, Schiller, Keats, Shelley and Byron. Byron was so moved by the independent spirit of Greek literature that he joined in the Greek war of independence against Turkey, where he lost his life (the Greeks still look on him as a hero). Shelley used the figure of Prometheus as a symbol of the independent artistic mind against the tyrannies of power and convention. Goethe also used Prometheus in a similar way, making man in his image (i.e. independent, and no slavish worshipper of a tyrannical deity).
In the 20th c., classical literature and its motifs have been revisited, often under the influence of psychoanalysis. So Freud and Jung look at Greek myths in the light of their own discoveries of the human mind, and novelists, poets and dramatists write versions of the ancient myths, Freud and Jung coming along for the ride. So Eugene O'Neill's Mourning Becomes Electra, though cast in a 19th c. American setting (New England after the Civil War), and a recasting of Aeschylus' Oresteia owes quite a bit to Freud. The brother and sister by the end, having killed their mother and her lover, have become their parents. James Joyce's Ulysses, which some would call the greatest work of 20th c. letters, takes the story of the Odyssey and recasts it as 24 hours in the life of Leopold Bloom, a Jewish advertising agent in Dublin, on 16 June 1904. Joyce has a little bit of everything in this work, but owes quite a bit to the classical training he received as a young man.
In painting and sculpture too, we see classical myth and history adopted as themes by many of the great artists of the modern period, but again, each generation takes and adapts the myth to the prevailing views of its time. So gods and goddesses often appear as lords and ladies in Renaissance painting, while Pablo Picasso takes the Minotaur (think of the importance of the bull to Spanish thought) and uses him as a symbol of the horror of war.