Today, in The Cambridge History of Judaism, Volume Three: The Early Roman Period, I read an essay by W.D. Davies and E.P. Sanders, "Jesus: from the Jewish point of view".
Davies and Sanders present Jesus as one who predicted the imminent kingdom of God: God's dramatic intervention into the present world order to create a new order. And yet, Davies and Sanders acknowledge that Jesus believed that his own ministry (e.g., exorcisms) reflected the in-breaking of God's kingdom. According to Davies and Sanders, Jesus was controversial among Jewish authorities because he offered a path to repentance that was not based on Torah-observance, but rather on being a disciple of Jesus. Davies and Sanders do not think that Jesus was anti-Torah, however, for Christians after Jesus' death were still divided on issues such as the observance of the Sabbath and Jewish dietary regulations, showing that Jesus himself did not settle those issues. Davies and Sanders put these different things together by saying that Jesus' approach to the Torah was not systematic.
Davies and Sanders present a view that the Romans put Jesus to death because Jesus caused a disturbance during the Passover, when Jesus cleansed the Temple, and the Passover was an especially sensitive time. For Davies and Sander, Jesus was not just protesting corruption when he cleansed the Temple, but he was also predicting the destruction of the Temple by challenging the sacrificial system, which depended on the Temple merchants (since there were times when people had to purchase an acceptable animal at the Temple, especially since their own animals may be unacceptable). Although the Romans got rid of Jesus, they did not deem his followers to be a threat, and so the Romans did not kill people in the Jesus movement.
The criterion is dissimilarity appears in this essay---that's the criterion stating that things Jesus says or does that fit neither Judaism nor Christianity are most likely from him. Jesus in Matthew 8:12ff. and Luke 9:59ff. tells a man who wants to bury his father before following Jesus to let the dead bury their dead, and to seek the kingdom of God. Davies and Sanders deem this to be authentically from Jesus, for Jesus was going against the importance that Jewish religion placed on the burial of one's parents.