What exactly did Apollo do with respect to the Sun?
My understanding is that Helios (a Titan) was actually the one who drove the Sun Chariot across the sky every day. But why, then, was Apollo the god of the Sun?
Am I correct in this understanding? Was Apollo (as a son of Zeus) sort of Helios's boss? He could ride the chariot if he wanted to, but he was also free to mate with Daphne, Cyrene and so forth, and otherwise meddle in human affairs, while Helios had to drive the bus every day? Or did it work another way?
Apollo didn't have any solar properties in Homeric times, and he and Helios were clearly distinct entities. Helios is extensively discussed in Book XII of the Odyssey, for example.
From the 5th century BCE and onwards Helios started to be identified with Apollo. An early reference to the fusion of the two beings can be found in fragment 781 of Euripidis' Faethon, where the poet tells us that Helios is "rightly called Apollo":
ὦ καλλιφεγγὲς Ἥλι᾿, ὥς μ᾿ ἀπώλεσας καὶ τόνδ᾿ Ἀπόλλων δ᾿ ἐν βροτοῖς ὀρθῶς καλῇ, ὅστις τὰ σιγῶντ᾿ ἀνόματ᾿ οἶδε δαιμόνων.
The association became a lot more commonplace during Hellenistic times.