The symbol on the left is the Cauduceus or a herald's staff. In Greek mythology, it is an attribute of Hermes, the God of messengers, economists, merchants and - incidentally - gamblers, liars and thieves. Hermes is the closest Olympian God to economics and business, that's why the Cauduceus shows up on this website - and in the seal of some economics schools, for example this one.

How the Cauduceus ended up as a symbol for medical institutions in the U.S., I don't know. Maybe it got confused with the Rod of Asclepios?