I think there are a lot of good answers already, so this is superfluous, but it sums up my sentiment:
If I have numerical references, I can easily skip them when reading a paper and I can easily find them in the bibliography. They are also easy to note down on a piece of paper.
It is easier to find reference 124 then to search for a name in a long list of authors... - Add to that, numerical lists are typically created by tools while some authors insist on handwriting author-year references. Great if the reference you are looking for doesn't exist...
The author-year reference then brings along the issues mentioned by other contributors: Does it become part of the text or it is a label? If I write it as if part of the text it impacts the writing style. Then how do I add a long list of authors?
And as others have pointed out, whom do you list? The first author? (The robust approach) or the group leader (recognizable name). My personal BibTeX libraries are internally author-name(-a/b/...) which works, but means you are potentially referencing a "nobody".
This then might even impact reader interaction where you will be more attentive to a recognizable reference than an unrecognizable one which is objectively obviously wrong.
So numerical references provide a wonderful opportunity to attribute work neutrally in the written manuscript.
And on a comment to other responses:
I agree that references without titles are problematic as most papers can be found with a title and an author, but random jumbled number with cryptic abbreviations are sometimes wrong and universally harder to find... (Even worse if the literature of interest spreads over a wide field of journals and sub-disciplines....)
Fortunately "digital paper" is cheap and modern papers are no longer impacted by the inherent cost of printing physical books, so this is less prevalent than in the past. (Though still an issue...)