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Their primary disagreement with the rest of the Church was over the treatment of those who forsook their faith during the Persecution (303 - 305 AD) of Diocletian. The rest of the Church was far more forgiving of these people than the Donatists were. As a result, many towns were divided between Donatist and non-Donatist congregations.
The bishop Augustine campaigned against this heterodox belief throughout his tenure as bishop of Hippo, and through his efforts the Church gained the upper hand. His successes were reversed when the Vandals conquered North Africa. Donatism survived the Vandal occupation and the Byzantine reconquest under Justinian. It is unknown how long this belief persisted into the Moslem period.