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)
Source: Al-Islam.org
