A duty, from the Webster's New Collegiate Dictionary, is an assigned service; conduct due to parents and superiors. An obligation, also from the Webster's New Collegiate Dictionary, is a duty imposed legally or socially; a thing that one is bound to do by contract, promise, moral responsibility.
To me, an obligation is something you are obliged to do. It is a responsibility. Something a person is bound to do ¦ a commitment or a requirement. In an obligation, there is no room for a choice, although choosing not to take responsibility can result in