To maintain residency, green card holders have to continue living in the US. They will lose their status as soon as they leave. They are not allowed to vote in general elections and some local elections and cannot vie for the office of the president. They will continue to remain citizens of their country thus are not allowed to own a US passport.
However, citizens of the US can live anywhere on earth for as long as they would like to and not lose their citizenship. They are allowed to vote and if they are citizens by birth, can contest for the office of the president. They are also allowed to own a US passport.
Matt Pickering, Adds
The privilege of voting, occupying offices which require US citizenship, becoming the US military officer, being allowed to own a US passport, not having to renew your citizenship after every ten years, not having to inform the USCIS every time you relocate and not being deported if specific crimes are committed.