I had the chance to play one hardcore character in Diablo 3, and I found a quest in the crypts that I haven't been able to find with normal characters. Are there quests unique to hardcore characters in Diablo 3 or are some quests simply very rare encounters?
If you're talking about the Jar of Souls or Matriarch's Bones quests that randomly appear in one of the incorrect defiled crypts, then no. These quests are simply random and can appear on any difficulty or hardcore setting.
It remains to be seen for the rest of the game, I don't know of any specific answers that have been given out by Blizzard staff mentioning hardcore-specific quests.