How did the Christian movement distinguish itself from Judaism? Can anyone explain to me how Christianity distanced itself from Judaism and set itself apart?
In 70 CE, Second Temple Judaism came to an abrupt and traumatic end, with the destruction of the Temple and the enslavement of many leading Jews. Randall Price says in The Secrets of the Dead Sea Scrolls, page 137 (citing Dr. Lawrence Schiffman):


  Second Temple Judaism can now be seen as a transition period in which the sectarianism and apocalypticism of the period gradually gave way to rabbinic Judaism, on the one hand, and Christianity, on the other.  Indeed, it is now clear that the Second Temple period was a kind of sorting out process.


John Dominic Crossan agrees, saying in The Birth of Christianity, page xxxiii, that it is not really accurate to say that Christianity eventually broke away from  Judaism. He goes on:


  It is more accurate to say that, out of that matrix of biblical Judaism and that maelstrom of late Second-Temple Judaism, two great traditions eventually emerged: Christianity and rabbinic Judaism. Each claimed exclusive continuity with the past, but in truth each was as great a leap and as valid a development from that common ancestry as was the other.  They are not child and parent; they are two children of the same mother.


Christianity and rabbinic Judaism were picking up the pieces in 70 CE and, although competitors for the hearts and minds of those who had followed the former Second Temple Judaism, it was in the interests of neither to attack the other. In its early days, Christianity did not choose to distinguish itself from rabbinic Judaism, with Christians regularly attending the synagogues set up and controlled by rabbinic Judaism. It was eventually rabbinic Judaism that initiated the break, banning Christians from the synagogues in the 90s of the first century, probably in response to the increasing anti-Jewish rhetoric of the Christian movement and the refusal of gentile Christians to undergo circumcision. John Shelby Spong says, in Born of a Woman, page 65, the liturgy of the synagogues was reformulated by 85 CE to attack heretics, and the Christian Jews were finally expelled.

Everett Ferguson says in Backgrounds of Early Christianity, page 461-2, Gemaliel's grandson, Rabban Gamaliel II (active 80-120), introduced into the Eighteen Benedictions, the curse which effectively excommunicated Christians: 


  “Let the Nazarenes and the heretics perish as in a moment, let them be blotted out of the book of the living and let them not be written with the righteous."