On this website, there are 120 questions tagged citation-style, and probably many more without the tag (a search for APA returns 244 results).
I work in applied mathematics, and among my colleagues the general approach to citation styles varies between "it doesn't matter as long as it's consistent" and "I use Bibtex and that's it; frankly I don't even know how this style is called".
Why is citation style considered important in other fields and receives so much interest? What is it that I am missing?
There are styles because otherwise it would be a complete mess. I speculate that it has to do with the way archiving and indexing of publications are done in a given library/institution/society, and the type of publications in the field (peer-reviewed articles vs book chapters): in some fields you may retrieve a publication with Journal + Volume + Page, whereas in other ones, you need Authors + Title + Year + Publisher. So different needs, different styles.