As they follow different doctrines, it is expected that the development of Christianity and Buddhism were fairly (but not entirely) separate, but their spread was nearly identical by traders and missionaries.
The foundations of Christianity are simple, but it would soon become a popular world religion. The religion follows the life of a teacher, Jesus of Nazareth, and the revelations he made. He shared his truths at the Sermon on the Mount. In time, disciples of Jesus came to think of him as a savior of the Jewish people. Christians would model the Buddhist monks and build monasteries at centers of learning. It initially appealed to slaves and women, the lower classes, until the bishop St. Augustine wrote The City of God and appealed with intelligence to the upper class.
Buddhism, like Christianity, arose from humble origins. In 534 B.C., a kshatriya prince named Siddhartha Gautama left the comforts of his home to find an explanation of suffering. Like Jesus, he proclaimed the fundamentals of Buddhist doctrine at Deer Park. Gautama became known as the “Enlightened One”, or Buddha. Later, he would be thought divine like Jesus Christ. Buddhist monks built monasteries wherever they came to teach their faith. And like Christians, Buddhists were consisted of members from the lower castes.
Spread of these world religions was similar in their carriers and limits. Christianity spread when the Roman emperor Constantine endorsed the Edict of Milan, allowing Christians to practice their faith openly in the Roman Empire. Later, the emperor Theodosius made Christianity the official religion of the empire. Quite differently, Buddhism was limited until the fall of Han China, when a replacement was needed for Confucian philosophy. Both religions were effectively spread by merchants in their trade. In Christianity, missionaries resembling Paul of Tarsus and Gregory the Wonderworker were themselves popular and effectively spread the religioin. In India, Ashoka sent his subordinates to Sri Lanka and other countries to encourage Buddhism in neighboring lands.