Civic duties and civic responsibilities are essentially the same. A civic duty is a responsibility that a citizen of a country has, so for all intents and purposes, the two terms can be used interchangeably. The working concept behind these terms is that citizens have obligations to serve their country, and in return, those citizens are given certain rights and/or protections.
Some civic duties are actually mandated, and citizens are required by law to do them. An example of a mandated American civic duty is jury duty. Another example of a mandatory civic duty is paying taxes. While people might constantly complain about this particular civic responsibility, those taxes do pay for things like firefighters, police, and even road maintenance. A third example of a civic duty that I myself participated in was registering for Selective Service. That was mandatory for me. An example of an optional civic duty is voting. I am allowed to vote; however, I don't have to vote either. I do so to be an active participant in my country's government.
To come up with examples of responsibilities in your private life, you'll need to think about the duties or obligations you have to yourself or to the people closest to you. Responsible activities I do in my private life include spending time with my kids, coaching, staying up to date with information that helps me do my job, and volunteering in my neighborhood.