I disagree with this. At best, this is a square-rectangle situation. I’d agree that many, maybe even most, stoner metal bands are also doom metal bands, but the vast majority of doom metal bands are not stoner metal. No savvy person would refer to Thou as stoner metal.
And counterintuitive as it sounds, what a band calls themselves actually does not matter at all. Genres are descriptive, not prescriptive; if Depeche Mode came out today and called themselves a black metal band it wouldn’t make their music not synthpop/dark wave.
Frankly, a lot. I’m a big fan of rateyourmusic, so I’d recommend starting there:
https://rateyourmusic.com/genre/black-metal/
https://rateyourmusic.com/genre/death-metal/
That being said, without being immersed in the genre it can be hard to tell the difference, and description alone can be unclear, so I’d really recommend listening to a few examples from each of those pages.
And it is worth noting that both of these are huge genres with a number of subgenre children below them, so some of this is going to be in broad strokes.
In general, black metal is going to be largely defined by consistent use of tremolo-picking and blast beats supported by double bass; death metal uses these elements but these are a consistent fixture of black metal. Black metal tends to be very “wall of sound” in its approach - everything all at once in maximalist ways - but with little to no strong regard for production or recording quality. In terms of pace, black metal tends to be almost punk-like in tempo, driving and fast but rarely breakneck-blistering. And the vocals of black metal tend towards high pitched shrieks and fry screams, though clean singing (especially theatrical) is not terribly uncommon (though an all clean singing black metal song is basically unheard of). Lyrically (side note: I am not a fan of defining music in genre terms by lyrical content), black metal has a long history of very particularly anti-Christian, pro-Satanic lyrics, and unfortunately there is also a large segment of black metal that is full of disgusting white supremacist rhetoric; thankfully that segment has largely self-segregated and you’re unlikely to run into them in most modern metal circles. And the lineage of black metal is strongly influenced by punk rock; black metal would not exist without punk.
In contrast, death metal may feature tremolo-picking and double-bass supported blast beats but these are usually textural parts of a greater whole. Death metal was born directly out of thrash metal (itself the evolution of short-lived speed metal), so death metal sounds a lot more “metal” than punk-influenced black metal. Beyond tremolo picking, guitars in DM tend to be very down-tuned and feature very prominent riffs; where black metal is “wall of sound,” death metal is frequently very very guitar and/or vocals-forward, and these elements usually sit very clearly in the mix rather than blending in. Songs are also traditionally more progressive in structure than black metal which tends towards repeating motifs. The tempo of death metal is often very fast, but this is not a requirement. Vocally, death metal is where you hear “growling” most. These vocals are usually low false chord growls rather than the high-pitched fry scream shrieking of black metal. Lyrically death metal can be extremely diverse with no real set expectations like in black metal, though a sense of the grotesque and macabre is certainly common. While not wholly detached from punk music (nothing post-speed metal really truly is), there is no glaringly obvious punk DNA like you can hear in black metal.
Keep in mind these are generalities and meant to apply to the “traditional” or “baseline” versions of these genres; the sub- and microgenre children thereof all have their own rules and whatnot that can radically change this.
Jon Barbas, music critic and reviewer from Heavy Metal Philosophy has a saying that I fully agree with: “Death metal has the highest floor but lowest ceiling in metal while black metal has the lowest floor and the highest ceiling.” What he means by that is that most black metal is genuine garbage but the good black metal is transcendent. Death metal has a super high average quality rating, so if you listen to any random DM band you’re probably gonna have a decent time; black metal has a super low average quality rating so if you just pick any random BM band you’re probably gonna have a terrible time. But man, if you take the time to find some good black metal, goddamn is it good.
All this said, I’m not an expert, just an avid fan whose “special interest” happens to be heavy music genres. I highly recommend you read the RYM pages and listen to some of the examples from the corresponding lists to get a feel for it yourself.
edit:
If you don’t want to read and just want some listening recommendations to hear the difference, check these out:
For Black Metal: My favorite modern black metal band is Gaerea, though their recent work has drifted away from black metal into being some kind of black metal metalcore hybrid (I love it). Go back to their 2022 album Mirage and listen to that, or at least the songs Salve and Deluge. And/or listen to the album In The Nightside Eclipse by Emperor from 1994, which is a classic of the genre - just know that it’s technically symphonic black metal so there will be additional symphonic elements (“pure” black metal, at least good pure black metal, is actually kind of rare - a lot of the quality stuff is a subgenre or strongly influenced by other genres).
For Death Metal: Absolutely start with 1989’s Altars of Madness by Morbid Angel. Not only is that a great example of the truly typical death metal sound, it’s also just one of the best albums in the whole of extreme metal. Beyond that, Cannibal Corpse is probably the next logical step, as by far the most successful DM band of all time - their song Hammer Smashed Face is probably the single best known death metal song in the world.