The two intertwined serpents-the oldest found in Sumer...representing the manipulations carried out on the inhabitants of the earth-with the Mesopotamian "Tree of Life" frequently shown, or as the Olmec "Tree of Life" (Mesoamerican Cosmology)-as in this link:
[link to www.world-mysteries.com
again at another link the pertinence of healers who are motivated by different things-one desiring to heal and help others heal themselves, and the opposing side, with 2 serpents specifically geared more to PROFIT
"Friedlander surveyed 242 logos or insignias of American organizations relating to health or medicine in which the caduceus or staff of Asclepius formed an integral part dating from the late 1970s to early 1980s. He found that professional associations were more likely to use the staff of Asclepius (62%) while commercial organizations were more likely to use the caduceus (76%).
The exception is for hospitals, where only 37% used a staff of Asclepius versus 63% for the caduceus [but remember that US hospitals are usually commercial ventures].
Friedlander notes that while the prevalent use of the caduceus for the commercial aspects of medicine might be seen as "more-or-less appropriate", he thinks the reason is that professional associations are more likely to have a real understanding of the two symbols, whereas commercial organizations are more likely to be concerned with the visual impact a symbol will have in selling their products."
"Friedlander, Walter J. The Golden Wand of Medicine: A History of the Caduceus Symbol in Medicine." New York, Greenwood, 1992
from a link posted earlier-
[link to drblayney.com
A book that elucidates and clearly explains the differences and they are considerable is this one(link below the quote)
"Unfortunately, although both the staff of Aesculapius and the cadeuceus are in use, only one of them has a historical connection with the practice of medicine.
The other is a symbol of certain qualities best left unassociated with the physician.
A look into the origins of these icons shows that the true symbol of the physician is the staff of Aesculapius.
The cadeuceus, with its wings and twin snakes, is an impostor that somehow managed to find a connection to medicine, a connection that is unfounded and undesirable."
[link to www.medspirit.com
It should help clarify-I do not like the double serpent one for medicine as it is an imposter and seeks to lock our expanding consciousness down IMHO