Actually it was the Swedish death metal scene that though Black metal was fag.
The second generation of black metal shared the same melody driven song structures as Swedish death metal at that time.
Concerning the part of share of both styles. Honestly, both are quite different, actually even black metal bands at the beginning where quite different among them.
Enslaved, Emperor, Hades (Almighty), Gorgoroth, Immortal, Mayhem, DarkThrone, etc. All those bands were quite different among them. Those bands were influenced by Helhammer, Celtic Frost and Bathory, among others of course, lets not forget Sarcofago from Brazil, and the rest of punk, thrash, and heavy metal influences.
if you take in consideration song structure you'll realize how different Death and Black are. Death being mostly a riff salad in which of course you need to have lots more of technique as a guitarist. In the other hand, in black metal you need to work a lot with the melody.
Concerning the ideology, Death was obsessed with the 'Only death is real' which is an old idea. Black was more fascinated with
fantasy, heroism, (both sharing a nihilist view though).
I'm not trying to say one style was better than the other. But both have a lot of differences which make them unique. (I love both styles and grew up with both of them).