Attorney and lawyer are interchangeable. They're the same.
It depends on what the grounds you're suing under as to what kind of attorney (or lawyer) you would need. Police misconduct? Malfeasance by city officials? Property rights? There are many potential areas of law that fall under this blanket of civil rights where you might end up suing the city.
No difference at all, check a thesaurus. For suing a city, it depends on the type of claim but any licensed attorney with experience in that field would be appropriate.