I don’t think that “bullet hell” differs from shmups so much as it is a subcategory of shmups. A “bullet hell” shmup is a specific type of shmup where there are an overwhelming number of shots on the screen to dodge.

I’m sure that it’d be perfectly-acceptable and appreciated to post here!

EDIT: Here, Wikipedia has a better description than I do:

https://en.wikipedia.org/wiki/Shoot_'em_up#Bullet_hell

Bullet hell (弾幕, danmaku, literally “barrage” or “bullet curtain”) is a subgenre of shooters in which the screen becomes crowded with complex “curtain fire” enemy patterns. It is also characterized by collision boxes that are smaller than the sprites themselves, to accommodate maneuvering through these crowded firing patterns.[29][30] This style of game, also known as “manic shooters”[7] or “maniac shooters”,[31] originated in the mid-1990s as an offshoot of scrolling shooters.[31] The DonPachi and Touhou Project series are early titles establishing the principle of bullet hells.[32]