The answer to your question lies in what each religion teaches about who God is.
From The Truth About Islam by Dr. David R. Reagan*:
God — The Koran asserts that the god of Islam is the God of Christians and Jews (Sura 29:46). Nothing could be farther from the truth.7
The god of Islam, Allah, is most definitely not the God of the Bible. Allah is presented in the Koran as an autocratic ruler who is aloof and arbitrary (Sura 5:40). Allah is unknowable whereas the God of the Bible is knowable (2 Timothy 1:12). Allah is impersonal, unlike the personal God the Scriptures reveal (1 Peter 5:6-7). Allah is unitarian (Sura 4:48) whereas the God of the Bible is trinitarian (2 Corinthians 13:14). Here is what the Koran says about the God of the Bible (Sura 4:171): “Believe in Allah and say not ‘Trinity.’ Cease! It is better for you! Allah is only One God. Far is it removed from his transcendent majesty that he should have a son.”
Allah is capricious (Sura 2:284), whereas the true God is trustworthy. And Allah is never anywhere presented as a god of love — which is the essence of the nature of the true God (1 John 4:7-16).
Jesus — The Koran denies point blank that Jesus was the Son of God (Sura 112:2-3). It also denies His atoning sacrifice by claiming that he never died (Sura 4:157). A substitute died for Him on the Cross. Jesus was translated to Heaven, like Enoch, where He will remain until He returns to kill all pigs, destroy all crosses, and convert the world to Islam. Jesus will marry, reign for 40 years and then die and be buried next to Muhammad in Medina.8 Jesus is characterized in the Koran as nothing more than “an apostle of Allah” (Sura 4:171).
The god that Islam teaches about is very different from the God of the Bible. Muslims and Christians may trace their roots back to Abraham but it's hard to claim they're the same god when they act so dramatically different.
Because Christianity stems directly out of Judaism, it's clear they are worshiping the same god. The God of the New Testament is the same as the God of the Old Testament. For example:
Jesus himself even acknowledges that salvation comes from the Jews (John 4:22). Christians may believe the Jews overlooked the Messiah but they undoubtedly worship the same God.
*More info about Dr. David R. Reagan and his denominational ties can be found here: