The Christian Perspective
Allah is a generic Arabic term for an ultimate creator God, and could be applied by Arabic speaking peoples to any ultimate creator God, whether He be the God of the Sikhs, the Muslims, the Christians, or the Jews; while Yahweh is the personal name of the God of Christians or Jews only (roughly translated "THE EXISTING ONE"); and Brahman is the non-personal God specific to Hinduism and related religions.
To most English-speaking westerners though, the term "Allah" is specifically synonymous with the Islamic or Muslim God, who is completely different from both the Christian God and the Hindu God.
Allah, the God of Muslims, has supposedly revealed Himself through the Qur'an. When the Qur'an was written, it claimed that:
A. the current Christian scriptures were true (Quran 2:40-44; 3:93; 5:43-48; 5:68; 7:157; 10:94; 20:133; 26:196-197; 29:46; 53:36);
B. the teachings of Moses, Abraham, and Jesus are true (Quran 3:84; 5:43-48; 42:13; 53:36);
C. Muslims and Christians worship the same God (Quran 29:46).
Indeed, according to the Qur'an, the Christian scriptures that were contemporary with it ought to be used to test the revelations given to Muhammad to see if they are genuinely from God (Quran 10:94).
Note that when these claims were made in the Qur'an (c. 600):
C. the God worshipped by the Christians was Trinitarian including the divine Son, Jesus (Isaiah 9:6, 48:16-17; Matthew 1:23, 3:16-17; John 1; Luke 9:35, 20:44; Acts 13:13; Psalm 110:1; Colossians 1:16, 2:9; Romans 15:16; 2 Corinthians 13:14; Nicene Creed c. 431);
B. Jesus had taught that God is Trinity (Matthew 28:18-20; John 14:9, 14:16-17, 14:26, 16:13), Jesus is divine (Matthew 28:18; Luke 19:38-40; John 8:58-59, 10:30-33, 14:6-7), and all men sin (Mark 10:18; Luke 11:13, 18:19);
A. the Christian scriptures (undebatably established since c. 397 in the west and c. 508 in the east, unchanged for 100 years prior to the Qur'an) taught that God is Trinity (Isaiah 9:6, 48:16-17; Matthew 1:23, 3:16-17; John 1; Luke 9:35, 20:44; Acts 13:13; Psalm 110:1; Colossians 1:16, 2:9; Romans 15:16; 2 Corinthians 13:14; Matthew 28:18-20; John 14:9, 14:16-17, 14:26, 16:13, etc), Jesus is divine (Matthew 28:18; Luke 19:38-40; John 8:58-59, 10:30-33, 14:6-7, etc), and all men sin as a direct result of Adam (Luke 11:13; Romans 5:12-21; 1 Corinthians 15:22, etc).
Note that Muslims deny the Trinity, the deity of Christ, and that all men sin as a direct result of Adam. The Qur'an also denies that Jesus was crucified (Quran 4:157), whereas the Christian scriptures that the Qur'an says are true claim that Jesus was crucified (Matthew 27:35, 28:5; Mark 15:24-25, 16:6; Luke 23:33; John 19:18).
From all of this we see that:
Premise 1: If the Qur'an is true, the Christian scriptures are true.
Premise 2: If the Christian scriptures are true, Islam is false.
Conclusion: If the Qur'an is true, Islam is false.
Additionally, it appears to me as though Allah claims in the Qur'an to always be just (Quran 4:40, also Quran 2:272, 2:281, 3:18, 3:108, etc), yet at times according to the Qur'an, He completely ignores the penalty for sin (Quran 4:64, 4:96, 4:100, 4:106, etc), thus making Him unjust. This makes Him both just and unjust at the same time and in the same way. In other words, Allah violates the logical law of non-contradiction. This means that either things can both be and not be at the same time and in the same way, making all knowledge utterly impossible, or else Allah as described in the Qur'an does not exist. Yahweh, on the other hand, does exist and tells us that it is impossible for Him to deny Himself, or to be self-contradictory (2 Timothy 2:13, Hebrews 6:18, Titus 1:2).
Brahman, the ultimate deity of Hindus, is also different from Yahweh. As I understand, under the Brahman deity, time has been going on for an infinite number of cycles, which means that since past time has ended, we have come to the end of infinity. Yet, infinity has no end by definition. Therefore, under Brahman, "infinite" and "not infinite" mean exactly the same thing. In other words, yet again, either Brahman does not exist, or else things can both be and not be at the same time and in the same way, defying the laws of logic and thus making all knowledge absolutely impossible.
Yahweh exists, whereas the other two do not, and only under Yahweh is it possible for reality to be as it is, whereas it would not be possible for reality to be as it is under Allah or under Brahman.