In the pathway to salvation, the core fundamental difference between Islam and Christianity is in the doctrine of "original sin" or the sin imputed upon mankind by the fall of Adam. The bible tells us that through the sin of one man, Adam, sin came into the world and mankind. The wage of sin is death. While each person is responsible for his own sin, the penalty or consequence of sin must be paid and do affect the progeny. Adam, the first man represents mankind and by his fall, the consequence of his sin is imputed unto man. Therefore, mankind in such a depraved state is not able to redeem himself unless God provides the redemption which requires a saviour. In contrast, as Islam does not recognize there is such an "original sin" that corrupts the whole of mankind, its pathway to salvation does not require a saviour because there is no penalty to be paid and atonement to be made.
This is the reason why Islam's way of salvation is through good deeds i.e. by works; whereas, Christians believe in the saving grace of God through the Messiah Jesus Christ whose atonement saved those who repent and believe in Him in accordance with the redemptive requirement of God.
The article below (in PDF ) discusses this issue from both the Christian and Islamic narratives.
1. PDF Article: The Doctrine of Original Sin - Comparision between Christianity and Islam
CA103-13.1 The Doctrine of Original Sin-Comparision R2
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2, PDF Article: The Dilemma of Original Sin
CA101 The Dilemma of Original Sin in Isl
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