Why did Roman authorities fear the early Christians?

There were lots of reasons! I think one of the biggest ones was that by following their own set of beliefs and laws, the Christians weren't following the Roman's beliefs and laws. Not only did this mean that in an already unstable empire (revolts and wars with other empires who were constantly trying to take pieces of their empire) more factions were being divided off, but the people whom they conquered were supposed to accept them and submit to their authority. Also, by worshipping God and making Him their true leader, they were choosing him over both the Emperor and the Emperor's chosen deities.
There were also a lot of misunderstandings about the Christian faith in the beginning. They were seen as a cult (think about how we perceive cults even today) with weird rules. The people didn't associate or participate in normal everyday activities and walked around calling each other 'brother' and 'sister' including their wives and fathers and then spoke about drinking their prophet's blood and eating his flesh... You can see how the rumours of Christians being incestuous cannibals were easily believed!