Why do whips hurt so much? What exactly is the mechanism that makes a whip deliver such a strong impact? Elasticity, torque, or pressure? Just hitting something with a plank doesn't deal nearly as much damage. What's the difference?
 A: 
Just hitting something with a plank doesn't deal nearly as much damage.

Generally, a plank can be much more damaging than a whip. Whips may look cooler, but that seems to be about it.
http://en.wikipedia.org/wiki/Whip#As_practical_weapons
Differences in pain need not reflect the amount of damage done (if we can even define that properly). Soft-tissue bruising, abrasions, lacerations, fractures, etc. probably all cause pain, but possibly (I don't know), the pain levels are not in that order. I would think that whips are good at bruising and lacerations, whereas planks potentially cover a more wide range of damage. (There is a lot of stuff on the internet about surprisingly painful paper cuts, which might also be related somehow.)
A: The reason, a Whip hurts so much is that the tip of whip moves extremely fast, causing the skin to tear. 
The reasoning behind this is easy to analyze from momentum conservation. Lets take a convenient approximation, that the mass per unit length($\rho$) does not vary through the length of the whip. This is not how real whips are, but it will not affect the conclusion very much.
Initially, the whole length of whip moves, say with velocity $v$, if you observe the motion of the whip closely, then you will see that as time elapses, the initial momentum is concentrated on a smaller and smaller section of the whip, while the rest is almost static.
Now if the whip length is $l$ then intial momentum is $\rho l v$. If we look at a snap shot at some later time, and say we observe that $l_0$ is the length of the whip currently in motion. 
Then by momentum conservation, 
$\rho v_0 l_0 = \rho v l$.
This implies that the velocity of the moving end, $v_0 =\frac{v l}{l_o}$
as time elapses $l_0 \to 0$, $v_0 \to \infty$. The tip is moving at a very large speed,  therefore it is capable is piercing. A thin tip makes the effect more dramatic (due to smaller $\rho$), but does not change the essential mechanism. 
You may enjoy this video 
A: The speed of the tip of the whip can exceed the speed of sound.  From Wiki:

The crack a whip makes is produced when a section of the whip moves
  faster than the speed of sound creating a small sonic boom. The
  creation of the sonic boom was confirmed by high-speed shadow
  photography in 1927.1
There are at least three "modes of motion" that can produce the
  necessary speed in a whip to cause it to crack. The three are: a half
  wave, a full wave and a loop. These names are indicative of the shape
  of the bends in the whip as it is thrown. In all three, the initial
  motion is applied to the handle, and the resultant shape moves down
  the whip's body to the tip. The high speed of the tip is explained by
  the law of the conservation of momentum. Since momentum is a vector,
  it has a direction, and does not pass through any bend that reverses
  the direction of movement in the body of the whip − such as the one
  that occurs when a half wave shape moves down a whip.
When a whip is thrown, the initial motion of the handle adds some
  amount of kinetic energy to the body of the whip. If the whip is going
  to crack, the handle movement must also produce one of the modes of
  motion that create a reversal of direction in the whip's movement. As
  the reversal of direction moves down the whip, the momentum and the
  kinetic energy in the whip, are concentrated in the segment of the
  whip between the tip and the moving bend. As the bend approaches the
  tip, the mass of the moving part approaches zero while the energy
  remains relatively constant. Since the momentum is the product of the
  mass and speed of the moving object, the smaller the mass, the higher
  the speed. Hence the end of the whip moves extremely fast, easily
  reaching the speed of sound.
Many published popular science explanations capitalize on the fact
  that the general shape of a whip is tapered: thick at the handle and
  very narrow at the tip, hence the decrease of the mass. While tapering
  does contribute the decreasing mass, it is not a deciding factor. Even
  "flat" un-tapered whips will crack. The actual decrease of the mass of
  the moving part occurs simply because the whip ends: the closer the
  moving bend is to the tip, the less mass is in the part that's moving
  in the given direction.

A: It doesn't  actualy hurt here is an article about the type of whips jockeys use:
 http://www.guardian.co.uk/sport/2011/oct/18/jockeys-whip-didnt-hurt
But I think you mean why does it seem to hurt.  As @Alfred said this is because it exeeds th speed of sound. Sounds are traveling vibrations in the form of pressure waves in an elastic medium. In gases, sound travels longitudinally at different speeds, mostly depending on the molecular mass and temperature of the gas, and pressure has little effect.
 wiki says:  
 Some common whips such as the bullwhip or sparewhip are able to move faster than sound: the tip of the whip breaks the sound barrier and causes a sharp crack—literally a sonic boom.
Sonic booms generate enormous amounts of sound energy, sounding much like an explosion. 
This is exactly the situation of the whip.
