The dominant view among Muslims, Sunni and Shi'i, is that the Qur'an we have today is the Qur'an of the Prophet.
Exceptions apply.
Broadly speaking, exceptions in the Sunni tradition involve things such as the tradition from Aishah that a goat ate some ayat of the Qur'an; and in the Shii tradition, some narrations implying otherwise. Whether or not one accepts these narrations as authentic, or how one interprets them, is a different matter. (For instance, some Shii will understand hadith which seem to add some words to the Qur'an as examples of in-line commentary that was given while the Qur'an was being quoted, not as different quotations of the Qur'an.)
On the other hand, some ayat of the Qur'an are often understood to mean that the book itself is protected by Allah.
In any case, the Imams from the Ahl al-Bayt (A) have instructed us to use the Qur'an as it is in our prayers.
Based on narrations, possibly, a difference between the compilation of Imam 'Ali and the current Qur'an is the order of the surahs. As for other views, this may vary.
A helpful book to read about this in more detail in English is Ayatollah Ma'rifat, Introduction to the Sciences of the Qur'an. (Do not be deceived by the title, it is a heavy read!)
If I may offer my personal view (twice in one day!), there is really no guarantee about anything that happened in the past, when we were not present to witness it. In fact, we often can't even guarantee the absolute truth of things that happen in our time. However, we can try our best to come to a probable truth, with the help of Allah, and that is all we are expected to do.
