Although Christians and Muslims who speak Arabic use the same name for God and may desire to worship the same deity, the character of who they worship is different. Unless God's character is irrelevant, then the Muslim God and the Judeo-Christian God must be different.
The prevailing myth that all Gods and religions are essentially the same prevents people from recognizing behavioral differences arising from different belief systems. That thinking error helps explain why popular culture overlooks mistreatment of women and religious minorities in Muslim majority countries. Also, fundamentalist Christians in America become more stigmatized and less free as violence in fundamentalist Islam becomes more exposed, because Christianity and Islam must be the same.
Here are two examples of differences between the Judeo-Christian and Muslim Gods:
The Judeo-Christian God is three persons in one essence, while the Muslim God is a single autonomous entity. The Judeo-Christian God follows standards for interpersonal love and accountability as part of his essence in eternity. However, morality in relationships is not an essential trait of the Muslim God because relationships do not exist within a singularity. The Judeo-Christian God is law-abiding in eternal essence, but the Muslim God has no one to offend before creation. Consequently, according to the Qur'an, Allah is "the best deceiver" (wikiislam). But according to the Bible, God may not lie (wikianswers).
The Judeo-Christian God created men *and women* in his own image and incarnates himself in flesh and blood to restore relationship between God and man. The Muslim God may not be likened to anything in creation and most certainly did not become flesh and blood. If both men and women are in God's image, then God has both a male and female nature, all people are sacred, and all people are equal before God; including non-Jews and non-Christians. In Judaism and Christianity, sanctity and equality flow from partaking in God's nature. However, in Islam, human worth flows from the way that God created and fated men and women and not from any part of God's nature. Consequently, men and women are not equal before the law in Islam. Neither are Muslims and non-Muslims.
Differences between God in the Bible and God in the Qur'an result in behavioral differences among people. People become like what they worship. Character differences between Gods matter. Denying that they do insults all for whom God's character is important.