I can't seem to fathom how I know people who like bands like Soilwork but can't appreciate Carcass or At The Gates, and people who like Black Metal, but not Mercyful Fate.
I would think that when you like a band, you should at least make yourself familiar with the bands that inspired their sound, as opposed to only bands that sound like the one you like. I'm right about this, right? Nobody's asking anyone to like Mercyful Fate (They should be required to, though), but only to appreciate them. I can't see how someone could Like Celtic Frost, Emperor, and Mayhem, and NOT like Mercyful Fate. It just doesn't flow logically.
I admit readily that I'm not a huge fan of Black Sabbath or Judas Priest outside of a few songs, but I'd be lying through my teeth if I said I didn't appreciate what they did or the bands they inspired.
as an Aside, I think that if people looked for new music vertically (Influences and successors) rather than horizontally (Bands that sound the same), people would generally have better taste.
For example, this isn't music, but it's the same thing. Most know I'm an artist, and I love comic books. I just recently got into the comic artist Tim Sale, and his whole approach has... almost want to say "revolutionized" my approach to comic art, but that's not really it. More like, discovering him just represents the capstone of my long evolution, and I have a very clear idea of where I want to go stylistically now. Hence, he's a definite 'inspiration.'
Well, I got a book published about him and his work (excellent, called Tim Sale: Black and White), and he talks about his inspirations. Some of them we shared, but some, like comics legend Jim Steranko, I had never been exposed to. So, off I go to learn more about Steranko and his work, most notably on Nick Fury: Agent of Shield, from the 60's. This guy revolutionized comic book art. I may never be as hugely inspired by him as I am by Tim sale, but he's managed to inspire me just like he did Tim nonetheless.
So yes, I think if someone is a true 'inspiration,' you owe it to yourself to check out their inspirations.