Black metal has a heavier focus on atmosphere. The vocals are more of a screech than a growl, and the guitars use a technique called tremelo picking where one note is picked as quickly as possible for several measures.
Death metal is typically more aggressive. The vocals are usually growled (cookie monster vocals), and there are more guitar solos. Death metal is closer to the early thrash of Metallica, Slayer, Megadeth and Anthrax with brutal vocals and a focus on death.
There really aren't that many well known black metal artists. Burzum, Darkthrone, Mayhem, Immolation and Emperor are the most famous. There are several lesser known black metal groups such as Negura Bunget, Summoning and the whole Les LÃ©gions Noires which includes MÃ¼tiilation and several other French bands. The black metal scene in this country was pretty weak. In many ways, this music is meant to help you look at the darkness within you and reject the status quo, particularly the Christian power structure. In other ways, it's meant to scare the shit out of you.
Death metal is far more popular today, and many of the early black metal groups now play death metal. Death metal is being split into subcategories these days such as technical death metal (Crytospsy, Nile, Pestilence), symphonic death metal (Therion, Epica, Haggard), and Gothenburg (or melodic) death metal (In Flames, At The Gates, Amon Amarth). Personally, I'm not wild about symphonic or melodic death metal. I'm partial to the original form of death metal (Atheist, Death, Morbid Angel, Immolation, Incantation, etc) and technical death metal.
Blackened death metal typically has the music of death metal and the vocals of black metal. Tremelo picking is used more than traditional death metal too. A lot of bands that started off as black metal, but changed to death metal fall into this category.
There are many other forms of metal such as traditional heavy metal, groove metal, speed metal, thrash, power metal, drone, doom, stoner metal, folk metal, viking metal, grindcore, deathcore, gorecore, mallcore, and so on.
It took me a long time to understand this. I was a heavy metal fan in the 1980s, but got sidetracked in college to indie stuff, and later grunge. I decided to get back to my metal roots because I was sick of all of that whiny crap that passes as metal today (Staind, Korn, Bush, etc), the garage scene (White Stripes, The Strokes, etc), today's "punk" (I use that term loosely) and everything that gets played on the radio. I was shocked to find out how much great metal came out while I wasn't paying attention. Whole genres have lived and died in that time, and I missed it.
I've gotten to be somewhat of a metal snob, but I still listen to most of my CDs, from Tool to The Pixies and from The Bad Brains to Lynyrd Skynyrd. There are a lot of things in metal that annoy me. I don't want to ever hear a pig squeal in another song. That has been completely overdone. Breakdowns are completely pointless too. For crying out loud, if you're going to do something like that, please do something more interesting than "chug chug, chug chug, chug chug" for a full minute. And if your production quality is shit, it better be due to the fact that you had no budget and were recording on a 4 track in your basement. It doesn't add atmosphere, it just sounds like crap. Overly dramatic, opera vocals suck too. If I wanted opera, I would listen to Rossini or Mussorgsky.