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.