"Prosecute" descibes an action taken by the authorities to institute criminal proceedings against the suspects for a crime. Civilians cannot prosecute anyone. Prosecution is done normally by a unit under an attorney general, which is a government post. "Prosecute" applies to criminal cases only. The Prosecution represents the Crown or the State. "Sue" is used in civil cases only. In a civil case, there is a plaintiff and one or more defendants. There is no Prosecution.

Sue - someone sues you for damages of some sort. They will take you to court. Prosecute - being prosecuted for a crime you committed. The government /police charge you for it whatever the crime is. I hope this helps.