A snake sheds its skin whole, emerging with the old one left behind intact. No other common animal offers so plain an image of renewal, and almost every positive association snakes carry traces back to it.
The medical emblem is the clearest case. The Rod of Asclepius is a single serpent coiled around a plain staff, carried by the Greek god of healing, whose sanctuaries at Epidaurus and elsewhere kept live non-venomous snakes; patients slept in the abaton and the snakes moved among them. It is the genuine and ancient symbol of medicine.
The caduceus is a different object: a winged staff with two serpents, carried by Hermes, god of commerce, thieves, travellers and messengers, with no medical association whatever. Its use as a medical badge is a nineteenth- and twentieth-century error that took hold particularly in the United States, decisively when the US Army Medical Corps adopted it in 1902. Professional medical bodies in most of the world use the single serpent; American commercial medicine is full of the wrong one. This is a well-documented mistake and it is worth naming precisely.
Scripture gives both valences. Genesis 3 makes the serpent the subtlest of creatures and the tempter, and the curse on it is aetiological, explaining why snakes move as they do. But at Numbers 21:6-9 Moses is instructed to make a bronze serpent and set it on a pole, and those bitten who look at it live. The object persists: 2 Kings 18:4 records Hezekiah destroying a bronze serpent called Nehushtan because Israel had been burning incense to it. John 3:14 turns the pole into a type of the crucifixion, so the same image serves as tempter and as saviour within one canon.