Oral tradition lasted much longer in Islam than in other religions, which means that despite the invention and adoption of writing, the Qur'an specifically wasn't written down for longer than other texts were. Because of this, the Qur'an skipped the scroll, and was began being written down in the codex format. It's likely that the text we know today as the Qur'an was standardized over roughly three centuries.
Calligraphy and the art of writing something down were considered sacred and for hundreds of years the manuscript was the only acceptable way to distribute the Qur'an. The first printed versions of it came out of Venice around 1537, almost 100 years after the first Christian Bible was printed. In addition to cultural limitations, however, the Arabic script did not easily give way to printing. With the many diacritics that come with Arabic, creating pieces of type for the language was much more complicated and time consuming than for that of the Roman alphabet.
There was a steady flow of Arabic works printed in western Europe from the 17th-19th centuries, but the first press that was established in the Arab world to produce Arabic works for Muslims wasn't established until the 19th century. And while printing Islamic works began in Europe in the 1530s, a Muslim wasn't actually involved in the production until 1787.
Islamic calligraphy and the Arabic language have had massive impacts on the world as we know it today. The accuracy of both global and regional maps in Arabic texts from as early as the 12th century were unmatched until modern times. Most surviving Islamic manuscripts date from the 16th century or later (the most I've encountered have been from the 19th century).
(Information from Geoffrey Roper in "The History of the Book in the Muslim World" from The Book: a Global History. Edited by Michael F. Suarez, S.J. and H.R. Woudhuysen. Oxford University Press: 2013.
