The answer may have as much to do with semantics as with theology, but semantics are important.
In essence, the religion of Islam began in 610 A.D. when a man named Mohammed became convinced that the polytheism practiced by the Arabian tribes was wrong, and that only one god should be worshiped. Mohammed believed the angel Gabriel revealed this to him, and preached the message widely, teaching his followers the "revelations" from Gabriel that were collected and preserved as the Qur'an.
"Allah" is a poetic form of the Arabic al illah, meaning "the god." Mohammed took an incipient belief in a supreme god and promoted Allah as the only god.
Mohammed and his followers identified Allah as the god of the Old Testament, consider Abraham to be their spiritual ancestor, and revere the biblical prophets.
Muslims also consider Jesus to be a miracle-working prophet who was born of a virgin. They consider it heresy, however, to claim that Jesus is the Son of God, and reject all notions of the Trinity.
Islamic arguments against Christianity typically assert that Christians worship three gods and thus show infidelity to the one god.
The core of Islamic faith is expressed in the shahadah, sometimes translated as "There is no god but Allah, and Mohammed is his prophet." Some English-speaking Muslims translate "There is no god but God."
Jews and Christians have more commonality in belief, and clearly refer to the same deity when we say "God." Christians believe, however, that God's self-revelation does not stop with the Old Testament but is fulfilled in the New Testament.
Whether one prefers to say that Muslims, Jews and Christians believe in different gods, as opposed to differing views of the same god, is largely a matter of semantics. From a Christian perspective, anyone who does not accept the full revelation of God through the saving work of Christ and the sustaining presence of the Holy Spirit has only a partial understanding of God.
Thus, Christians might argue that Muslims or Jews worship "a different god" because we believe their concept of God is incomplete. It is perhaps more appropriate, however, to think of Muslims and Jews as worshiping the same god, though not in His fullness.
Why does it matter? Our terminology can impact the effectiveness of our witness to any who do not accept Christ. It is essential that we keep channels of communication open by showing respect for people of other faiths, even if we believe their view of God is inadequate. Explaining Christ as the saving fulfillment and ultimate revelation of the same god is a natural and effective means of sharing our faith with Muslims and Jews. Insisting that they worship a different god altogether is bound to be counter-productive.
It is possible to be tactful in our speech without compromising our witness.