Before listing off the reasons why Jesuits were despised by their competitors and opposition, I'd like to note that going into the early seventeenth century, the Jesuits actually enjoyed a pretty respectable reputation among lay people and potential priests. The Spiritual Exercises had proven to be very popular among even lay readers, and the Jesuit emphasis on education and the study of natural philosophy/science alongside theology gave them a notoriety as studious, diligent missionaries, particularly among Catholics in France and the Netherlands. Not only that, but their accomodation strategies across the world, both in Latin America and Asia, had proven very successful in converting formerly "pagan" societies to Catholicism. Moreover, ethnographic and scientific information from these exotic places found their way back to European audiences. To give an example, the mid-seventeenth century Jesuit scientist/historian Athanasius Kircher's most famous work, China Illustrata, borrowed from secondary accounts of China and the Chinese that previous Jesuits had reported; Kircher himself never set foot in Asia, and the accuracy of his work suffers for it as well, but nonetheless, his works required several printing runs and translations, demonstrating its popularity.

I bring this up not just as a corrective, but also to show one reason why the Jesuits were disliked or at least treated with suspicion by their colleagues. Over their first hundred years in practice, the Jesuits had become extremely successful, not only in combatting Calvinism and converting heretics around the world, but also in terms of gaining the esteem of princes and other wealthy patrons.

Competing for resources was obviously a major sticking point for the other orders who needed funding for their own missions (the Dominicans and the Franciscans had already established residences in China and Japan, after all), but moreover the way that the Jesuits found success raised some eyebrows. In the mission fields, the Jesuits practiced a very liberal and accomodating method of converting people. They would translate liturgies, primary texts, and even the Bible to local languages, which might seem fairly innocuous, but the theological ramifications behind adopting the Chinese word "shangdi" to refer to God as opposed to the Latin "Deus" were considered severe among contemporaries. Moreover, they would also adopt certain rites and try to appropriate ideas and concepts from local religions to make Catholicism seem much less foreign. On top of that, while the Franciscans and the Dominicans converted from the bottom-up (that is, teaching the commoners and then forcing a conversion of society from the masses) the Jesuits worked from the top-down, converting the nobility of an area who would then do the work in supplanting local religions for them. In order to appeal to the elites, this often led to Jesuits like Matteo Ricci for example, adopting the traditional garb of a Confucian scholar, and leading a less poverty-stricken life than their colleagues often did. All the while, they were speaking the local language and possibly practicing or allowing the practice of local rituals. Yes, this made them very successful, but to their Dominican and Franciscan confreres, was this the type of success they wanted, or were the Jesuits inadvertently condemning the souls of locals to Hell with slipshod conversions?

Or, some even ventured, were they even trying to convert people to the Catholic church, or were the Jesuits simply trying to seek power and fame for their own ends? This is probably the criticism most people are most familiar with when it comes to the Jesuits; they were king-makers, conspirators, and power-hungry priests who stuck around at court to try and overthrow the government. In places like France, especially right after the French Wars of Religion, any sort of destabilizing factor was immediately distrusted. Their direct supervision from the Pope made them even more suspicious in the minds of Protestants like the Dutch, who were currently in revolt against Spain, English, whom Spain had recently attempted to invade, and certain Germans, who were in the midst of the Thirty Years' War.

And to be fair, Jesuits had gotten involved in some conspiratorial activity. A famous English recusant named William Allen, for example, who was not a Jesuit but was deeply involved in the establishment of the Jesuit English college at Douai, encouraged, supported and even agitated for an invasion of England by the King of Spain. Real or imagined, however, this became the reputation that clinged to them throughout the seventeenth and eighteenth centuries, where we see the Jesuits being expelled from courts throughout Europe. While the events that lead to the suppression of the Order in 1773 are extremely complex and have many different motives, both the suspicion of political ambitions, and several controversies over rites and their accomodative practices were major influences in why they were disdained.

(Some) Sources:

Donald Mungello, Curious Land: Jesuit Accomodation and the Origins of Sinology

Lynn A. Martin, The Jesuit Mind: The Mentality of an Elite in Early Modern France

John O'Malley: The First Jesuits

James Broderick, The Progress of the Jesuits

George Minamiki, The Chinese Rites Controversy: From Its Beginning to the Modern Day.

edits to fix formatting

u/NicLewisSLU has covered much of what I wanted to say, but I would also add that - as argued by several historians who consider such changes from the perspective of empire - hostility to the Jesuits was also augmented by their existence as a trans-imperial group. Appreciating this helps us understand why the Jesuits were expelled throughout imperial domains specifically in the mid-eighteenth century.

This was a period which saw profound efforts at centralisation among leading western imperial powers - for Spain, as one example, efforts to impose imperial authority were prompted following the War of the Spanish Succession, and in particular following defeat at Havana (1762) in the Seven Years’ War. Similar trends can be seen in Portuguese imperial endeavours (Pombal’s reforms especially).

Neither was centralisation unique to the Iberian empires; the British and French empires are further obvious examples. Imposing imperial authority and exploiting colonies more than ever before was necessitated by the experience of intermittent but frequent colonial warfare, the costs of which only increased through the century.

The presence of missionaries who were not obviously subordinate to respective imperial hierarchies posed an obvious threat to imperial authority. There is little coincidence, then, that the expulsion of the Jesuits from Spanish and Portuguese imperial domains coincided with a period of imperial centralisation. The Jesuits, then, were not just a domestic threat. Their presence was also a threat to the conduct of empire. Similar trends can be seen with other groups and bodies which may be considered transimperial. Crackdowns on contraband traders and interimperial trade, key measures in the Bourbon and Pombaline Reforms (Spain and Portugal, respectively), are one example.

There was obvious pressure from rival missionary groupings and from others in society; but it’s important to consider their role beyond the domestic. The fact that the Jesuits posed a threat to the imposition of imperial authority is also commonly cited as a major factor in their change of fortune in the eighteenth century.

Edit: u/NicLewisSLU mentioned Matteo Ricci as an example of Jesuits departing from “orthodox” missionary practice - you might also enjoy the fact that Ricci literally avoided mentioning the crucifixion in his literature to elites, because he feared it would anger them. It’s hard to consider any greater departure from one of the most central tenets of Christian theology. This might have been exceptional, but the Jesuits really did go to some interesting ends in order to convert.