Basically, persecution is an unjust punishment or cruel and unjust treatment of an individual or group of individuals usually based on race, religion, income level, sexual orientation or other beliefs. In other words, some people are treated in a cruel and unjust way just because of a certain category into which they happen to fit.
Prejudice is the belief that these individuals or groups of individuals somehow deserve this treatment. They have a bias against this group based upon nothing more than the fact that this group is different.
Those that carry these prejudices tend to afflict most of the pain and torment upon these "different" groups. Because prejudice comes from a place of fear, it is logical that these individuals who persecute others do so out of fear - fear of losing their social standing, their money or their jobs. They blame those they persecute for any and all of their problems even though there is no evidence to support this. As long as someone is "beneath" them, they are in less danger of falling lower on the social or economic spectrum, and they will do anything to keep them there.
In this novel, the poor, white families carried such prejudice against the blacks. The poor whites fear that the blacks might, just might, surpass them on the social and economic scale. For this reason, it is the poor whites, such as the Ewells, who are desperate to keep the blacks down, even if this means making up lies and ultimately contributing to one's death.