Qur'an Compilation

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. 

It is narrated that Imam Ali (A) collected the Qur'an after the death of the Prophet (S) but his collection was not accepted, and it was inherited by the Imams and is with the Mahdi (A), along with some other inherited sacred objects. 

Instead, the collection we have today is the one endorsed by Uthman during his caliphate, and the Shia Imams also instructed their followers to use that, for instance, in salat. 

No one can really say how much the compilation collected by Imam Ali (A) might have been identical to or different from what we have now, except that it is said that some the surahs were in a different order, since his was arranged chronologically, so this might at least be a difference. 

This is what one can glean from narrations - of course, not every narration is accurate - Allah knows best. In any case, we have been instructed to use what we have. 

Here are some references you can follow up: https://en.wikishia.net/view/Mushaf_of_Imam_Ali_(a)

Insofar as we do not physically have the Qur'anic manuscript compiled by Imam 'Ali, it is not possible to answer this question with the certainty that comes from seeing things with one's own eyes. Rather, it is necessary to rely on historical reports, and the validity of historical reports can be debated.

Anyway, one difference is that it is reported that Imam 'Ali added commentary and explanations to the codex that he compiled, including an explanation of when/why the ayat were revealed (asbab al-nuzul) and also a broader explanation of the ayat.

It is also reported that Imam 'Ali's manuscript was arranged chronologically, in order of the date of revelation, whereas the 'Uthmanic codex is not arranged chronologically.

In any case, we have been told by the Imams to use the codex that we have today as it is, so it is good to have faith that this is what we are meant to be using.

Allah knows best.