The main definitions of prosecute are (1) to initiate legal proceedings against, (2) to carry on or engage in, and (3) to pursue an undertaking to completion. Persecute means to oppress or harass, especially because of race, religion, sex, or sexual orientation.
The adjectives are occasionally confused because of their similarity in sound, but their roots differ and their meanings have no common ground. If you have trouble keeping them straight, just remember that prosecution is a legal procedure (in its most commonly used sense), while persecution is not.
Melbourne’s legal system will increasingly be used to prosecute accused people smugglers intercepted off the north coast of Australia. [The Age]
In fact, as the Star reported some time back, Christians are the most persecuted people around the world. [Toronto Star]
[B]ut what is wrong with saying that we need to prosecute the war on terror more intelligently? [The Atlantic]
One of the worst things a community can do is persecute their resident artists. [Meadow Lake Progress]
The UN body was set up to prosecute serious crimes committed during the wars in the former Yugoslavia. [BBC News]
Belgium was occupied by Nazi Germany from 1940 to 1944, and there was active collaboration with the Nazis in persecuting Jews. [Associated Press]