ELI5 Why Drop Shadows look so different Illustrator vs Photoshop

Well, my position on this might prove unpopular, but to put it bluntly, drop shadows don't belong in Illustrator.

Generally speaking, I'd contend that raster-based effects (and/or vector-editor trickery that should be raster based for best results), are consumer-level-market-driven add-ons to vector editing applications. I don't really have any sort of inside information about this, but I do seem to recall the introduction of drop shadow effects in Illustrator, and I remember that CorelDRAW! had the feature first, a few years prior. Considering that, it's reasonable to characterize the addition to Illustrator as a competitive maneuver (as opposed to anything resembling a 'good idea').

Perhaps my line of thinking is out of fashion, especially among users who would habitually complain about having to move among several applications to achieve their objectives, but actually, your post kind of justifies it. If good-looking, well-controlled drop shadow effects are needed in your output, your output (at least the shadows) should be created or applied in a raster-editing environment. All my work in Illustrator, Photoshop, and InDesign is strictly segregated this way according to the nature of its constructs, using the application originally designed to facilitate those constructs, and it doesn't limit me in any way. All the cross-functionality (often wrongly perceived as 'integration') that has come about among these applications as they've evolved is only bloat that was never actually needed, in my opinion.