When you get down to the fundamentals of theology, it ain't easy to be a real Christian, that is, a real follower of the teachings of christ without believing that some of the beliefs of Judaeism are wrong... although most people think that they can do this just fine.

1) Jesus is the Messiah -- take it or leave it this is what separates the christians from the jews on the most basic level (there are of course cultural separations but im going home soon so im not going into that) -- if you're christian, you beleive that Jews are wrong in thinking the messiah has not yet come.

2) Jesus' actual speech and teachings fly in the face of conventional Jewish teaching. Jesus spoke of the unimportance of ceremony as opposed to belief -- i.e. the following of the outward forms of a religion as opposed to the spirit of that religion (this is shown, among other places, with all that circumcision of the heart being more important than circumcision of the flesh buisness). Judaeism of the time, and still much of it is today, concerned with Tradition and ritual. The fact that some sects of Christianity have also become obsessed with Form over Substance is an accident of history and no fault of Jesus'.

All of this aside, however, I would watch how you use the word Anti-Semitism -- anti-semitism is often used in the context of being against the Jewish people rather than the Jewish theology and carries a fierce negative connotation ever since the second world war. It also implies an active hatred rather than a passive disbelief.