No they do not tend to differenciate between the 2 groups. Here are some examples that prove that they do not.
1) In April of 1978, Israel passed a law that stated that any christian who tried to convert a Jew to Christianity could face up to 5 years in prison. And asserted that an act as simple as giving a bible to someone could be considered trying to convert them. At that time parliment minister Binyamin Halevy called christian missionaries a "cancer in the body of the nation."
2) Israeli scholar Yehoshafat Harkabi wrote that he was disturbed by “these manifestations of hostility-the designation of Christians as idolaters, the demand to invoke the ‘resident alien’ ordinances, and the burning of the New Testament.”
3) Desicration of Christian property has long been a problem in Israel. Including the buring of the New Testiment and arson and vandalism to churches.
4) A catholic priest told reporters in 1948 “Jewish soldiers broke down the doors of my church and robbed many precious and sacred objects. Then they threw the statues of Christ down into a nearby garden.”
5) In 1948 the Christian Union of Palestine complained that 10 churches in Jerusalam were used by the Israeli army as bases and were desecrated by the Israeli army. Due to this 3 priests were killed.
6) Another such instance happened in 1967 Reverand James Kelso of the United Presbyterian church said “How did Israel respect church property in the fighting...? They shot up the Episcopal Cathedral [in Jerusalem], just as they had done in 1948. They smashed down the Episcopal school for boys...The Israelis wrecked and looted the YMCA...They wrecked the big Lutheran hospital...The Lutheran center for cripples also suffered...”
&) In 1963 hundreds of Jews(not the Israeli military) attacked Christian missions in Jerusalam, Haifa,and Jaffa. In the Church of Scottland school in Jaffa children were beaten when over 200 Jews attacked it.
A Palestinian-American Christian lawyer, when visiting Isreal or Palestine was strip searched several times even as a child. Since she was Christian and a child I highly dout that she posed any real threat of terrorism so they just did that to discourage her and her family from coming back. Please watch the video here to see her testimony and that of other victims. http://www.ifamericansknew.org/about_us/easiesttargets.html
In addition to that there is a Christian woman living in the Palestinian territory and she says that the Christians there are treated just as badly as the Muslims to them a Palestinian is a Palestinian.