Obviously both of these religions share some common threads. However, there are enough differences to be noticeable during their beginnings. Both religions share an origin that centers on a central character. This would obviously be Jesus of Nazareth in the case of Christianity and Siddhartha Gautama in the case of Buddhism. Both religions spread slowly initially but picked up speed after the death of their founder. A big contrast between the two religions is related to the locations that they developed in. In the case of Buddhism, northern India and a large part of mainland China had largely regional religious beliefs. Even in the areas that were given to practicing Hinduism, there was a great deal of variety in the religious concepts; even with regard to deities. This allowed for an easier spread of Buddhism initially.
With regard to Christianity, it began during Roman times. The Romans had strict laws regarding allegiances to Rome and the Roman government. The fact that christians identified themselves first by their religion as opposed to the government created a problem for early Christianity. It was seen as subversive cult and was banned from spreading in the open.