Why did the Romans respect Judaism?
The Romans respected Judaism because it was considered one of the ancient religions, and because the Jews had significant clout with Rome. Rome also recognized and protected the vast sums of money Jews across the empire sent to the Temple, never allowing any of it to be lost.
Les explains that the Judaistic religious people had clout with Rome, and the Romans had a lot of respect for Judaism because it was considered one of the ancient religions. He also notes that Jews from one end of the Roman Empire to the other would send vast amounts of money to the Temple as offerings, and Rome recognized and protected it, so that never was a dollar lost. This respect and clout allowed Jews to demand extradition of Jews they wanted to deal with in their own religion, bringing them back to Jerusalem. That is why Saul had authority to go to Damascus, outside Israel, and Rome permitted him to arrest Jews and drag them back to Jerusalem. By contrast, the Romans persecuted the Christian group because it was something totally new and was treated like a sect, unlike ancient Judaism.