You should first note that there were several Islamic Empires, the two largest being the Umayyad and Abbasid Dynasties. The areas of Eurasia and Africa which became Islamic were known as the Dar Al Islam, the House of Islam. The Dar Al Islam occupied large portions of Spain and Northern Africa, and the Middle East. Other than Spain, Western Europe was largely unaffected. The Mongol Empire under Ghengis Khan was the largest the world has known, occupying China, the Middle East, and large portions of Russia to the edge of Western Europe. It's advance was only stopped by the death of Ogedai Khan. Trade was protected in both, although in the Islamic Empire, most trade was with Islamic Merchants. The Mongols were tolerant of religion, and took no position on religious practice. Islamic kingdoms did "protect" those who were considered "People of the Book," that is Christians and Jews, but they were forced to pay a tax. No other religions were tolerated in Islam.
Prior to the creation or the founding of the Mongol states, the region in question belonged mainly to the Islamic empire, which was under the control of the Abbasids (northeastern Africa, Arabia, and the near East) and the Ayyubids (in Syria and other areas) until the siege of Baghdad by the Mongols in 1258. There are many similarities and differences between the Mongol states and those that existed before their astonishingly rapid and successful invasion of the region. First of all, geographically speaking, whereas Baghdad was seen to be the center of the Islamic empire for 5 years, the Ilkhanids’ capital was Azerbaijan. Despite these differences, the Mongols, and more specifically, the Ilkhanids, shared many similarities with those who existed before them. For example, under Hulegu’s rule, there was a period of significant intellectual expansion; educated men from many different regions were welcomed in the Ilkhanid court where they exchanged information with Muslim scholars. This kind of intellectual exchange was also seen during the Islamic caliphate, more specifically under the rule of Harun al-Rashid, which was considered to be the Golden Age of Islam. Furthermore, the Mongols were originally a nomadic people who achieved a great amount of success in their conquests of the region. The Muslims were also originally nomadic, and throughout the course of their history, they achieved remarkable success both in geographical and social terms. Although the Mongols were initially believers in paganism and Buddhism, many of them actually converted to Islam, starting with the Golden Horde. This element can be seen as both a similarity and a difference between the Mongols and the Muslims; it is a similarity in the sense that the two empires eventually practiced the same religion but the important difference to note here is that Mongol’s accepted the religion of the people they conquered, whereas the Muslims imposed their religion on the people they conquered.