Why do some fields care so much more about citation styles?
The purpose of a citation style is to provide the reader with references in a standard and legible way. Using a certain style within a research field makes reading and skimming through citations within the field's publications more efficient.

Uniform citation style within a unit of certain field makes the unit's publications look uniform and adds to the quality of the publication, just as good grammar and otherwise good appearance does.

Arguably, a big fuss is when students are learning the correct ways of citation.