As far as I understand, the critical theological questions as between the rationalist schools are this: 1) does god principally operate the universe through a system of natural laws, or does god directly will all aspects of the environment; and 2) to the extent god operates through natural laws, can study of those laws reveal divine truths[1].

I don't think Ash'arism denies natural laws--and may not even categorically reject any salience to faith in their study--but it disfavored and de-emphasized the importance of natural laws, significantly diminishing impetus for systematic support by the state, society, and religion for the study of natural philosophy. Institutional support for ethics, jurisprudence, logic, etc continued. Study of natural philosophy also continued, but it was more fragmented and shifted to the periphery of the Islamic world. Forms of Mu'tazilism survived at the periphery, for example in Al-Andalus (Moorish Spain).

It wasn't purely a religious issue; it was also a secular cultural issue. Followers of Mu'tazilism (who were largely the political and religious elite) tended to see themselves and their empires as inheritors of Greek culture, at least in terms of esteem, which is one reason why they were so eager to preserve, study, and extend Greek texts. But most people had negative perceptions about the Greeks and Greek culture, and to the extent Greek culture was associated with materialism and natural philosophy, they tended to turn away from and look for alternatives to those areas of inquiry.

[1] This is a big factor in the Galileo Affair, and why debate about astronomical models carried such weight and seriousness--because the ruling religious authorities were invested in the notion that divine truths were revealed through natural laws. Therefore, misunderstanding of natural laws invited heresy.