Public speaking is the process of communicating information to an audience. It is usually done before a large audience, like in school, the workplace and even in our personal lives. The benefits of knowing how to communicate to an audience include sharpening critical thinking and verbal/non-verbal communication skills.
A debate is a structured argument. Two sides speak alternately for and against a particular contention usually based on a topical issue. Unlike the arguments you might have with your family or friends however, each person is allocated a time they are allowed to speak for and any interjections are carefully controlled.
A good debater never deviates from the topic. He/she knows the topic thoroughly and has the ability to clearly place points and express everything.