Many people believe that Allah and Yahweh are the same God though they have very different characteristics that I don't think they're the same. However, since the Arabs are Ishmaelites they obviously know God. Could it be that Allah is a partial interpretation of Yahweh that came about because the founding fathers of Islam didn't have a full understanding of Yahweh, or is Allah an altogether different entity?
Just because a group descends from an individual, it doesn't follow that they are as knowledgeable as that person. Adam knew God personally, but just seven generations later the world was so wicked, God destroyed it in a flood.
Mohammad was familiar with Judaism and Christianity before he set about creating his own religion. In reading the Qur'an, you can see how the Bible influenced his teachings, but it is also clear that he did not have a good recollection of the Bible, so many of the stories are altered. Take a look at Answering a Muslim Critic and A Study of Islam for more details.
Just because someone claims to have a new message from God, it doesn't make it so, nor does it mean that God is revealing Himself in different ways. Allah is roughly based on the God of the Bible, but Allah is not the God of the Bible.