Technically speaking, Muslims do not worship a moon-god. Their proper name for God may have originally came from the name of their moon god, but Muslims are quite clear that God is transcendant and very far away from material existance. In fact, it could be argued that the God of Islam is TOO transendant- he has supposedly never even spoken to men directly. Men are not even created in God's image, because God is unlike anything in material existence, as well as completely different from all creation. This is the main reason that the God of Islam is not the God of Judaism and Christianity- it is a far better argument than the "Islam's God was taken from an ancient Arabic pagan moon God" argument.
Instead, I would argue that the reason why Islam's God is different is, as I mentioned above, he is too transendant. The God of Judaism interacts directly with the people- the God of Islam only sends angels. The God of Judaism created men in his image- the God of Islam only created men as servant to be his viceregents on earth (you can ask any Muslim, and they will tell you- men have nothing in common with their God). The God of Islam is different from the God of Christianity for the same reasons, and more. The God of Christianity became man, fully sharing in human nature (one man, two complete natures, two wills- the human fully in conformity to the divine). The God of Christianity is three persons who share in one essence and hence are one God, while the God of Islam, while being just one God, is also only one person. This list is by no means complete, but I just wanted to point out that the arguments that Islam's God is not the same as the Judeo-Christian God should be taken in the opposite direction from the way I see it going- they are different in their very natures.