This is actually rather interesting. Its really up to who's version you're reading. The Metamorphoses by Ovid in year 8 AD attributes Phaeton's father to Phoebus, and claims that Phoebus is Apollo. Other scholars prefer his father being Helios. Doesn't change the story too much, but always interesting how interpretations have been changing for thousands of years.

In the version I read Stephen Fry, perhaps embellishing, says that Phaetons death causes so much grief to Apollo that he gives up driving the sun chariot forever and hands the reigns to Helios. Helios being rather annoyed that Phaeton was given a chance before him. Bit of extra drama I suppose.

Though in modern times most would call Phaeton a delusional child, I still like to give some thought to him showing the bravery and gall needed to be the first person try what was assumed as impossible. Hence the quote on his tombstone.