The story is almost the same with some incidents and names being different. For example, Biblical story calls the parents of mankind Adam and Eve while Quran calls them Aadam (Arabic) and Hawwa.
(Courtesy: About Islam)
The incidents are almost the same in both Quranic and Biblical stories of creation but that does not mean that Muslims and Christians pray same the same God. According to the story of creation in both Bible and Quran, Adam was created in his own shape by God and then Eve was created as Adam’s companion, out of his own rib.
But the difference occurs in the reception of the story. While Christians call God as the father of Adam and Eve, Muslims refuse to see Allah as father and takes him only as the creator of the mankind. So both Islam and Christianity might be following the same story of origin, but their reception is quite different when it comes to the theological aspects of both the religions.
Another point of difference occurs when we talk about the real son of god, Jesus, who according to Christianity, took birth on Earth to sacrifice himself for the Original Sin. Had the story of Adam and Eve been exactly the same in Islam and Christianity, it would not have been mentioned in Quran that the worshippers of Jesus shall go to hell.
One more Biblical concept that Quran or Islam vehemently opposes is that of Trinity. According to Islam, there is only and only one god.
So, the stories of Adam and Eve might look similar superficially, but when it comes to the respective religions, there are many diversions and differences in the two.