In Vampire Diaries S02E21 and also a few near-by episodes, it is said that Klaus is cursed - as he is both a Vampire and a Werewolf and more powerful than both.
How is it a curse and not a blessing?
It's a curse because he has the disadvantages of both. He thirsts for blood while suffering the effects of turning into a Wherewolf.
From Vampire Diaries Wiki:
This lack of control and their susceptibility to mind-control is a curse which only hybrids have.
EDIT: In light of OP's edit, narrowing down the discussion to Klaus only, the possible explanation is the worst of both the species - Vampires and Werewolves. The thirst for blood, as well as the pain of turning into a wolf.