What is the difference between (and importance of) extrinsic vs. intrinsic rewards in gamification? What is the definition  of extrinsic and intrinsic rewards?

Why are they important to, and what is their role in, gamification?
Intrinsic Rewards (motivation)
These are rewards which are directly of value to the person, rather being something that represents (or could be "traded in") for something to of value.

So getting a badge in and of itself isn't an Intrinsic Reward, but up votes on an answer would be b/c it converts Status and Validation of the user's work.
These rewards are also intrinsic to the task. They have not been "bolted on" or "dangled like a carrot" in front of the user.

Extrinsic Rewards (or incentives) are rewards which  are not naturally a part of the task.
So if you enjoy drawing and I compliment you on the drawing that's intrinsica. If I offer to pay you to draw more, that's extrinsic. 


from this slideshow.

Extrinsic Incentives are demotivating and, thus, should be avoided in Gamifying a system
There is a lot of research that shows that Extrinsic Rewards are demotivating because:
They extinguish internal motivation


Result in Habituation, requiring higher levels of the External reward
to achieve the same effects.
Send a message that you think they should not want to do this (why
else would you need to bribe them)
Seem manipulative


Therefore, Extrinsic Rewards should be avoided.
And if you find that your "players" are focused more on the game mechanics than the work then that's a clue that your game mechanics may be offering extrinsic motivation.