They are similar in a basic outline, but different on a few important points.
First, let us consider the basic outline. Adam and Eve were placed in a garden in a state of happiness. God instructed them to eat from any tree except one tree. Eventually, they were deceived into eating from the forbidden tree and their nakedness became apparent to them. God then expelled them from the garden.
Now, the disagreements.
1. Neither book names the fruit, but the Bible alone calls it the tree of knowledge of good and evil. The Qur'an does not call it such, but teaches that humans are already inspired with the knowledge of good and evil at creation in order to enable them to exercise choice between good and evil. This knowledge did not come as a result of eating from a forbidden tree.
2. The Bible says the deceiver was a serpent, but the Qur'an says it was Satan.
3. The Bible says that Adam was not deceived, but only Eve was deceived; it says that Eve then gave the fruit to Adam and he ate. On the other hand, more than one Qur'anic passages mention that they were both deceived. One passage specifically says that Satan approached Adam and deceived him. The Qur'an does not single out Eve for blame in any passage.
4. The Bible says that when the couple heard the sound of God walking in the garden in the cool of the day, they hid from him among the trees. So God called out to Adam asking where he was, and asking if he ate from the forbidden tree. On the other hand the Qur'an does not depict God in limited human form. The Qur'an and the Bible both teach that God knows everything always.
5. According to the Bible, when the couple was confronted with their mistake, they blamed each other, and Adam even blamed God because God gave him the woman who gave him the fruit. According to the Qur'an they did not pass the blame. Instead, both repented.
6. According to the Bible, God cursed them. According to the Qur'an, God forgave them and guided them.
7. According to the Bible, they were driven out of the garden because God was afraid that they may eat from the tree of life and live forever. According to the Qur'an, God's plan was to educate our first parents in paradise, then send them into the world for a limited time to resist Satan, the enemy. They were sent to earth as part of God's plan for them; not as a way of preventing them access to the tree of life, but as a test to distinguish those deserving of everlasting enjoyment in God's paradise.
8. According to the Bible, God had said that when Adam eats from the tree he would surely die, and the serpent said they will not surely die. The serpent was right - they did not die. Contrary to this, in the Qur'an, God said that if Adam and Eve eat from the tree they will become wrongdoers, then they will have to leave the garden and come out to where they will have to labour. Satan, however, promised them that if they eat from the tree they will live forever. Satan was wrong - they did not live forever.
9. According to the Bible, because of God's curse, serpents have to crawl and eat dust, women have to suffer in childbirth, and men have to sweat for a living. According to the Qur'an, no such curse was issued. The difficulties of life on earth are what makes it different from life in paradise.