Why is Apollo's symbol a snake?

It isn't. Apolla's symbol is the sun, the harp, and golren arrows.

Hermes' symbol is the snake wrapped around a winged stick.

The snake is a common symbol for wisdom.

Apollo's son Asclepius has the symbol of a single snake wrapped around a staff (the symbol seen on most ambulances). Asclepius is technically the god of medicine, but Apollo frequently takes on the role of god of healing himself. Snakes are seen as a symbol of rejuvenation because of their ability to shed their skin, and many people believe that in ancient Greece, snake venom was prescribed as a therapy.