In a Nutshell:
The stories of the prophets that a related by the Qur'an are usually similar to and more often than not those in the Bible, both the new and old testaments.
Where They Agree
The general narratives are agreed in stories of:
Where They Disagree
Whilst there is a lot of agreement in overall narratives, they often differ on minor or important points.
Islam for instance rejects the stories of David as an adulterer (2 Sam 11, 21) as well as other prophets where the Bible made such claims (Gen 19: 30-38, 34, 38, 49, etc.).
The Qur'an also categorically condemns the notion of polytheism allegedly commited by the Prophet Solomon when he was deceived before his death (1 kings 11). In this case the Quran says:
وَاتَّبَعُوا مَا تَتْلُو الشَّيَاطِينُ عَلَىٰ مُلْكِ سُلَيْمَانَ ۖ وَمَا كَفَرَ سُلَيْمَانُ وَلَٰكِنَّ الشَّيَاطِينَ كَفَرُوا
"And they followed [instead] what the devils had recited during the reign of Solomon. It was not Solomon who disbelieved, but the devils disbelieved" (Surat al-Baqarah 2: 102.)
Likewise Muslims don't accept Jesus (Isa) was crucified instead believing that someone who resembled him was crucified and God saved him.
Whilst the general narratives are similar between the two revelations, details often differ, sometimes significantly.
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