Christianity did not easily spread through the Roman Empire. See the related question, "How did Christianity spread throughout the Roman Empire?"
It was Chrisitanity
Through most of their history the ancient Romans had their own religion: Roman Religion. In the Later Empire Christianity spread in the Roman Empire, was endorsed by the Roman emperors and became state religion.
Christianity
No religion helped to save Roman culture. One religion spread through the Roman Empire in its later days and transformed Roman culture: Christianity.
Catholicism
There are two "Islams". The first is the religion which spread primarily through peaceful interaction or the benefits it provided under the stewardship of an Islamic State. The second Islam was an actual Empire or Islamic State which expanded through warfare and conquest. It is necessary to understand that while the Islamic Empire grew through warfare, most conversions to Islam were not compelled.
The Roman Empire.
Religion spread out throughout the Mauryan Empire. Ashoka was the first ruler to promote Buddhism. He sent teachers to spread the religion throughout India. Buddhist teachings and the laws of Ashoka were carved on rocks and tall pillars for all the people to read. It influenced the Mauryan Empire that much.
Through truth.
Ashoka was important because he spread Buddhist religion to his empire.
The Islamic empire expanded through conquest. Islam as a religion spread both due to the imperial growth and due to peaceful conversions of those outside of the empire.