Death Metal is usually faster, has growls, where doom metal is very dark and usually doesnt have the growling vocals, frequently its also got slower power cords.
My favorite Death Metal is Opeth's first 6 Albums though that band is not really a purely death metal band as they play everything from Death Metal to Jazz/Funk ect...
Death Metal examples
Opeth -their old stuff (new stuff is totally different
Death
Morbid Angel (I dont listen to but they are OG death metal)
Doom Metal examples are:
Cirith Ungol
Saint Vitus
Then there is Black Metal which is about witchcraft and devil worship ect that has a lot of blast beats and a screaming/growling and solos (Behemoth is an example but I stay away from the overtly Evil/Lusciferian musci)