The caduceus is often used as a symbol of medicine and/or medical practice primarily due to widespread confusion with the traditional medical symbol, the rod of Asclepius, which has only a single snake and no wings.
To find the caduceus, with its two snakes and wings, used as a symbol of medicine instead of the correct rod of Asclepius, with only a single snake, is extremely common. This usage was popularised as a result of the adoption of the caduceus as its insignia by the US Army medical corps in 1902 .
The rod with two snakes is a Caduceus (3 of the shown) and the rod with one snake is a Rod of Asclepius (Bottom left corner).
The caduceus is a herald's staff, a symbolic object representing Hermes (or the Roman Mercury), and by extension trades, occupations or undertakings associated with the god. It is today typically depicted as a short staff entwined by two serpents in the form of a double helix, and is sometimes surmounted by wings. This staff was also borne by Iris, the messenger of Hera.
The rod of Asclepius is an ancient symbol associated with astrology, the Greek God Asclepius, and with medicine and healing. It consists of a serpent entwined around a staff. The name of the symbol derives from its early and widespread association with Asclepius, the son of Apollo, who was a practitioner of medicine in ancient Greek mythology.
Symbolism (From Wikipedia)
The serpent and the staff appear to have been separate symbols that were combined at some point in the development of the Asclepian cult. The significance of the serpent has been interpreted in many ways; sometimes the shedding of skin and renewal is emphasized as symbolizing rejuvenation, while other assessments center on the serpent as a symbol that unites and expresses the dual nature of the work of the physician, who deals with life and death, sickness and health. The ambiguity of the serpent as a symbol, and the contradictions it is thought to represent, reflect the ambiguity of the use of drugs, which can help or harm, as reflected in the meaning of the term pharmakon, which meant "drug", "medicine" and "poison" in ancient Greek; we know that today antidotes and vaccines are often compounded from precisely the thing that caused the poisoning or illness. Products deriving from the bodies of snakes were known to have medicinal properties in ancient times, and in ancient Greece, at least some were aware that snake venom that might be fatal if it entered the bloodstream could often be imbibed. Snake venom appears to have been 'prescribed' in some cases as a form of therapy.
The staff has also been variously interpreted. One view is that it, like the serpent, "conveyed notions of resurrection and healing", while another (not necessarily incompatible) is that the staff was a walking stick associated with itinerant physicians. Cornutus, a philosopher probably active in the first century CE, in the Theologiae Graecae Compendium (Ch. 33) offers a view of the significance of both snake and staff that is worth quoting at length:
“ Asclepius derived his name from healing soothingly and from deferring the withering that comes with death. For this reason, therefore, they give him a serpent as an attribute, indicating that those who avail themselves of medical science undergo a process similar to the serpent in that they, as it were, grow young again after illnesses and slough off old age; also because the serpent is a sign of attention, much of which is required in medical treatments. The staff also seems to be a symbol of some similar thing. For by means of this it is set before our minds that unless we are supported by such inventions as these, in so far as falling continually into sickness is concerned, stumbling along we would fall even sooner than necessary. ”—Asclepius: A Collection and Interpretation of the Testimonies, Baltimore, 1945
In any case the two symbols certainly merged in antiquity as representations of the snake coiled about the staff are common. It has been claimed that the snake wrapped around the staff was a species of rat snake, Elaphe longissima.