There is no ranking system to speak of for doctors. As Estelle mentioned, seniority and the connections made over the years are often what determine the opportunities that are open to you.
I would say that the biggest privilege of being a physician is one that all share: having the opportunity to care for people. Beyond this, it depends how you define "rank". Personally, the experience that I have garnered practicing medicine for over 30 years has earned me respect among my peers and colleagues. If you are talking about a hospital, then there is a hierarchy of authority that will define your rank within a hospital. The residents (new doctors) will be at the bottom of this hierarchy, working long hours and obeying the orders of the attending physician. The attending physician will supervise these residents, often working fewer hours. Then you have management positions (like a department head) in which you work with the hospital administration to ensure the hospital is capable of caring for all its patients.
This will probably depend on where they practice. Bigger cities and bigger hospitals will definitely have more perks than a small community and small town with the same titles. Everything would depend on the amount of money the hospital has.