>>12622534they both represent the same concept: a rod and the serpent, the serpent represents chaos or evil the rod represents the balanced center this idea is even more emphatic on the left figure where the rod even has wings this means that it is "higher" or "God-like" so the winged-rod represents good (or goodness).
all of this is the same concept as the staff of Moses, the story goes that people were bitten by serpents God commanded Moses to put his staff on the ground and a golden serpent appeared swirling on the staff and all who get close and saw it would heal from the poison
it was chosen to represent the medical field on this same basis, medicine uses poisonous sustances that when used right or on the right quantity they can heal
so the intrinsic meaning is: bad (chaos or evil) is turned around and used for good.
another curiosity is the greek term "Pharmakeia" which also is related to a poisonous sustance or drug and to sorcery or magic
https://www.etymonline.com/word/pharmacy https://www.biblestudytools.com/lexicons/greek/nas/pharmakeia.html 
