How was the Muslim world different than the European world during the early Middle Ages?

Answer 1

In some respects it was much more advanced, because learning, especially in science and maths, flourished under early Islam.

At the same time, learning in Western Europe stagnated into dogmatic, non-analytical repetition of Ancient Greek ideas.

Answer 2

By and large, the fundamental difference between the Islamic World (referred to as Dar al-Islam in the question) and Christendom (referred to as Western Europe in the question) between 750-1200 C.E. is that science, technological progress, and some types of pluralism were embraced in the Islamic World while unchanging religious theocracy was dominant in Christendom.

There are a number of reasons for this, but the dominant one is that the ruling dynasties in the Islamic World until the mid-1100s were made of Mu'tazilites, who were a sect of Islam (now extinct) that prized rationalist discourse over more traditional Sunnis and Shiites who prized orthodoxy in religion over rationality and used violence to maintain that possession. When the Sunnis and Shiites were able to overcome Mu'tazilite repression, they, in turn, repressed the Mu'tazilites and led to a rapid decline in scientific and technological achievement and a narrowing of the previous tolerance. Conversely, Europe was held in the grasp of the Unified and later Catholic Church which used its political power to completely repress any dissent. Europe began to improve in the late 1200s as more kings began to directly challenge the terrestrial authority of the Catholic Church, with the Wars of Religion of the 1500s allowing Europe to become more diverse and secular.