How serious of a security threat is the “Bash bug?”

I’ve heard reports of a new security threat called the Bash bug (or Shellshock), and some of them claim that this is a more serious issue than Heartbleed. Is this just hyperbole, or is the Bash bug really that great of a concern? If so, what makes it such a significant threat?

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Story added 25. September 2014, content source with full text you can find at link above.