Why many Republicans have a deep seated suspicion of expertise and science methods

Believe it or not, science is science, not political or a weapon for evil, unless it has been politicized or weaponized for evil. When politics is ripped out of the picture, science is neutral because reality just is what it is. Science doesn’t care what anyone believes the world or humans to be. However, many or maybe most Republicans no longer trust science or scientists, particularly when the science is politically or personally inconvenient. That distrust exists despite vast, undeniable benefits to modern life that come from from modern science. So why distrust science or scientists, and why is it the decline in trust more pronounced with Republicans than with Democrats?

Well, for the most part the answer to that is decades of demagoguery and propaganda from crackpots, rigid ideologues, radical right authoritarians and parties with lots of money at stake. They have intentionally and knowingly fomented irrational skepticism in science. Some science about this helps clarify the situation. First, the difference in trust is mostly partisan. Republicans really are more skeptical. In view of the overwhelming benefits of modern science, one can reasonably believe the distrust is mostly unwarranted (irrational), political and grounded mostly in decades of mendacity, i.e., the moral rot of demagoguery, propaganda, lies, deceit, etc. 

Second, solid evidence indicates that this partisan political divide is rooted in anti-regulatory, anti-government rhetoric, especially from the Reagan era forward. Conservative and authoritarian elites and institutions cynically framed inconvenient scientific findings, e.g., global warming, as ideologically hostile, politically motivated and thus not trustworthy.

Finally, research on right-wing authoritarianism and social dominance orientation predict rejection of science. Right-wing authoritarianism (a personality trait) and related ideological attitudes predict disagreement with scientific consensus in several domains. That arises from generalized but irrational distrust of science and scientists

It's hard to believe, but stuff like this still needs to be said in public.