To persecute someone is to treat them unfairly or in an oppressive way over a period of time because of their race, religion or political beliefs.
“Jews were persecuted in Nazi Germany during the 1930s.”
To prosecute someone is to press criminal charges against someone in a court of law.
“Illegal file sharers will be prosecuted under new Government plans.”
Note: The nouns persecution and prosecution have the same meanings as above but in the noun form. However, the term the prosecution refers to the group of lawyers in a trial representing the Government who try to prove that the accused is guilty of the crime.