I have researched this after a colleague brought it to my attention that it is not necessarily a Cadeceus. According to the AVMA the symbol is the Staff Aesculapius as stated in the following:
"The official seal of the Association Comprising the letter V super-imposed on the staff of Aesculapius encircled by the words American Veterinary Medical Association, and with a line encircling all of those elements, shall be registered with the U.S. Patent Office as a registered mark and shall be used by no other organization or individual without written permission of the Association" (AVMA Bylaws, Article X).This symbol was approved by the AVMA in June 1970 as the associations trademark. It was not until November 21, 1972 that the symbol was officially a registered trademark as decided by the USPTO. The AVMA has the following copyrighted as their logos to denote membership to the association:
Either with the circle and AVMA around the symbol or without.
Now from the research I have done there is no clear distinction that the Veterinary symbol of the single snake and staff is strictly for animal medicine. I have seen it noted as the symbol of medicine in general but most veterinary sites, schools, and hospitals agree that their symbol is the Staff of Aesculapius rather than Caduceus.
Now the Caduceus is generally a staff with 2 snakes and wings. This arose from the Greek God Hermes the Messenger which later turned into the Roman God Mercury. The staff was attributed to Hermes from the story of Tiresias:
In "Les Mamelles de Tiresias" (The Breasts of Tiresias) tells how Tiresias--the seer who was so unhelpful to Oepidus and Family- found two snakes copulating, and to separate them stuck his staff between them. Immediately he was turned into a woman, and remained so for seven years, until he was able to repeat his action, and change back to male. The transformative power in this story, strong enough to completely reverse even physical polarities of male and female, comes from the union of the two serpents, passed on by the wand. Tiresias' staff, complete with serpents, was later passed on to Hermes...This magic used to transform Tiresias was also seen later on in time in the use of alchemy. Alchemy had some connections to medicine, chemistry, and metallurgy which were attributed to Hermes (Mercury) and described using the image of his two snaked staff.
Staff of Aesculapius
The staff with one snake arose again from the Greek God Aesculapius. He was most likely a skilled physician who practised medicine in Greece around 1200BC (even described in Homer's Iliad). Through myth and legend he came to be worshipped as Asclepius, the Greek God of Healing. Now the origin of the staff has a much more grotesque beginning.
In ancient times infection by parasitic worms was common. The filarial worm Dracunculus medinensis aka "the fiery serpent", aka "the dragon of Medina" aka "the guinea worm" crawled around the victim's body, just under the skin. Physicians treated this infection by cutting a slit in the patient's skin, just in front of the worm's path. As the worm crawled out the cut, the physician carefully wound the pest around a stick until the entire animal had been removed. It is believed that because this type of infection was so common, physicians advertised their services by displaying a sign with the worm on a stick.So the single snake staff could have been started from a worm underneath the skin. Stiff think thats a better symbol than changing from male to female and back like some kind of Mrs. Doubtfire action.
Let the debate rage on but I am taking the side of Staff of Aesculapius.
Roman sculpture of Aesculapius with single snake staff.