In earlier Greek mythology, the sun was personified as a deity called Hêlios (Ήλιος, Greek for "the sun"), whom Homer equates with the sun titan Hyperion. Other sources say Helios is Hyperion's son by his sister Theia. Helios was seen driving a fiery chariot across the sky. He has two sisters, the moon goddess Selene and the dawn goddess Eos. Many think that Apollo becomes the Olympian "sun god", but this idea is mostly based on speculation and assumption. The equivalent of Helios in Roman mythology is Sol. See also Sol Invictus.
The etymology of Helios, unlike most of the major figures in Greek myth is Indo-European (Burkert p 17)
The best known story involving Helios is that of his son Phaeton, who drove the sun chariot to his own disaster.
Helios was sometimes referred to with the epithet Helios Panoptes ("the all-seeing").
The names of the horses were Pyrois, Eos, Aethon and Phlegon.
Helios was worshipped throughout the Peloponnesus, and especially on Rhodes (an island he pulled out of the sea), where annual gymnastic tournaments were held in his honor. The Colossus of Rhodes was dedicated to him.
Helios was often depicted as a haloed youth in a chariot, wearing a cloak and with a globe and a whip. Roosters and eagles were associated with him.
"You will now come to the Thrinacian island, and here you will see many herds of cattle and flocks of sheep belonging to the sun-god- seven herds of cattle and seven flocks of sheep, with fifty head in each flock. They do not breed, nor do they become fewer in number, and they are tended by the goddesses Phaethusa and Lampetia, who are children of the sun-god Hyperion by Neaera. Their mother when she had borne them and had done suckling them sent them to the Thrinacian island, which was a long way off, to live there and look after their father's flocks and herds."
There were kept the sacred red Cattle of the Sun. Though Odysseus warned his men not to, they impiously killed and ate some of the cattle. The guardians of the island, Helios' daughters, told their father. Helios destroyed the ship and all the men save Odysseus.
Heracles and the Cup of Helios
While Heracles traveled to Erytheia to retrieve the cattle of Geryon, he crossed the Libyan desert and was so frustrated at the heat that he shot an arrow at Helios, the sun. Helios begged him to stop and Heracles demanded the golden cup which Helios used to sail across the sea every night, from the west to the east. Heracles used this golden cup to reach Erytheia.
Helios' Roman equivalent was Sol. On the Quirinalis, he was worshipped as Sol Indiges. The Circus Maximus housed another temple.
Emperor Heliogabalus imported Sol Invictus ("the invincible sun") from Syria.
A sun-god syncretism of Apollo and Mithras also referred to Sol Invictus was designated the god of the Roman Empire.
Helios and Apollo
Apollo as he appears in Homer, a plague-dealing god with a silver (not golden) bow has no solar features. But by Hellenistic times Apollo had become closely connected with the sun religiously. His epithet Phoebus 'shining' was later applied by Latin poets to the sun-god Sol also, perhaps from such connections as well as from its obvious appropriateness.
The earliest certain reference to Apollo being sometimes identified with the sun god appears in the surviving fragments of Euripides' play Phaethon. The play as a whole seems to have kept Helios separate from Apollo but in a speech near the end (fr 781 N²), Clymene, Phaethon's mother, laments that Helios has destroyed her child, that Helios whom men rightly call Apollo (the name Apollo here understood to mean Apollon 'Destroyer').
The identification became a commonplace in philosophic texts and appears in the writing of Parmenides, Empedocles, Plutarch and Crates of Thebes among other as well as appearing in some Orphic texts. Pseudo-Eratosthenes writes about Orpheus in Catasterismi, section 24:
But having gone down into Hades because of his wife and seeing what sort of things were there, he did not continue to worship Dionysus, become of whom he was famous, but he thought Helios to be the greatest of the gods, Helios whom he also addressed as Apollo. Rousing himself up each night toward dawn and climbing the mountain called Pangaion he would await the sun's rising, so that he might see it first. Therefore Dionysus, being angry with him, sent the Bassarides, as Aeschylus the tragedian says; they tore him apart and scattered the limbs.
Dionysus and Asclepius are sometimes also identified with this Apollo Helios.
But in mythological texts Apollo and Helios are almost universally kept distinct. The sun-god, the son of Hyperion, with his sun chariot, though often called Phoebus is not called Apollo except in purposeful non-traditional identifications. Roman poets often referred to the sun god as Titan.
It seems to be a modern meta-myth that literary references to Phoebus and his car or to Phoebus and his chariot refer to Phoebus Apollo in the role of sun god, rather than to Helios/Sol.
Saint Elias like Helios with his chariot
Nicolas Poussin. Helios and Phaeton with Saturn and the Four Seasons. c. 1629-30. Gemäldegalerie, Berlin, Germany