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Another question entitled "Do Christians and Muslims worship the same God?" requested the basis for the belief that Christians and Muslims do worship the same god.
In this question, I am requesting the basis for the belief that Christians and Muslims do not worship the same god, but worship different gods.
This question is specifically addressed to Christians who believe that Islam worships a distinctly different god, and that Judaism worships the same God as Christianity.
The answer to your question lies in what each religion teaches about who God is.
Islam
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.
Judaism
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: http://christinprophecy.org/?staff=dr-david-r-reagan http://christinprophecy.org/about/#Purpose
The God of the Christians/Jews is known as Yahweh (A.K.A. YHWH (or YHVH) Yahweh, Yehovah, Jehovah, and the LORD). The god of the Muslims is known as Allah. Jesus says in the Bible, the Holy Book of the Christians, in John 14:6 He is THE Way THE Truth and THE Life and no one comes to God (Yahweh) except through Him. He is the only way to God, He is the only true Savior, and the Life He gives us is True Life. Only through JESUS can you be saved and go to heaven, not through other religions. The Muslim god is therefore not the God of the Bible. No other god can get you to heaven, because they are not real. How can Allah get someone to heaven? He can't because he doesn't exist or is a demon (Deuteronomy 32:17).
For clarification: Jesus is believed by the Christians to be God in the flesh. Jesus is according to the Christians (of which I am a part of) believe Jesus is the Messiah of the Jews. Even if the Jews deny Jesus as the Messiah, Jesus is still the Messiah. God is actually a general term for a supreme deity. Whenever I use the word "God" ("G" capitalized) I am referring to YHWH. Whenever I use the word "god" ("g" not capitalized) I am referring to any false god. Allah is Arabic for God (https://translate.google.com/#en/ar/God) and Elohiym is Hebrew for God (http://www.biblestudytools.com/lexicons/hebrew/kjv/elohiym.html) although today used as a by-word or interjection (https://translate.google.com/#iw/en/%D7%90%D7%9C%D7%95%D7%94%D7%99%D7%9D).
