As duty refers to moral commitment. In the case of duty, the person will be involved in activity without any self-interest, such as your duties as a parent 作为父母的责任。

your duty for your country 对国家的责任Responsibility can be termed as an ability to act at one’s own will, without any supervision. It is the obligation to successfully complete an assigned task, such as your responsibility to clean to house 打扫房子的职责Obligation is the state of being forced to do sth because it is your duty, or because of a law, etc. such as your obligation to protect the environment 保护环境的责任。

contractual obligations 合同中所要履行的职责liability is the state of being legally responsible for sth。