This is mostly a copy of the much better articles:
https://yang-song.github.io/blog/2021/score/
and
https://lilianweng.github.io/posts/2021-07-11-diffusion-mode...
https://yang-song.github.io/blog/2021/score/
and
https://lilianweng.github.io/posts/2021-07-11-diffusion-mode...
I did reference the second article, although the score-matching connection to diffusion models was not observed by Song but instead by Ho et at. I didn't venture heavily into the score-matching aspect for the very reason you cited - Song's article on the subject is second to none!