The snake is from the rod of Asclepius, a Greek god of healing. He's actually a constellation because Zeus got pissed that he was too good at his job and could nearly raise the dead, so as with many other people who angered him, he stuck him in the sky. There's also a story that says he may have been Imhotep from way back in egypt. He was supposedly one of the first people to document cases, and when the greeks visited they were so impressed with his medicinal skills they basically diefied him. The huge zodiac debacle around new years was actually about adding his constellation to to cycle (though it wasn't really a new idea).
As for the winged one, it's technically the scepter of Hermes, and is thought to have been mistakenly used in place of the rod of Asclepius in the early 1900s, but you know, everyone likes wings and stuff that looks angelic, so it stuck.
