I recently heard that Shia and Sunni have different Qurans. Is that true? If so, why do they have different Qurans (that is different physical copies)?
This is not true, the Quran is the same and this is the opinion of most of Shia scholars.
The small kid who memorizes the Quran in Iran (i.e. Shia) will recite the same thing as another one who memorized the Quran in a Sunni country.
The major difference between Sunnah and Shia in terms of the sources are related to Sunnah or the Prophet -peace upon him- (i.e. the Sunnah).
For example, Shia does not accept the trusted hadith books of Sunnah as Al-Bukhari.