In the case of board breaking by a karate practicioner there are two possible scenarios. First is that the board isn't broken so the karateka's hand really hurts. In the second possibility the board is broken and the hand hurts a lot less. So, by the pain on the hand, we can assume that in the first case the force exerted by the board towards the hand is greater than the second case. Accordingly to Newton's third law, this force is the same with that of the hand's exerted to the board in each case. So, a relatively low force can break the board and a higher one can't?
Try it yourself - you don't need a board, you can just use a wall and a piece of paper. Let's assume that your punch is of standard strength. If you punch the wall, chances are you'll hurt a lot. If you punch the piece of paper, chances are you won't hurt at all. Since your punch is of standard and unchanging strength, why is there a difference?
The reason is the acceleration your hand feels at the moment of impact. When you punch the paper, the paper deforms around your arm before breaking. That means your arm takes (comparatively) more time to decelerate - say, 0.1s. When you punch the wall, the wall doesn't budge and your arm takes much less time to decelerate - say, 0.001s. That translates to an acceleration that's 100 times larger. By $F = ma$, the force you feel in the second case is also a hundred times larger than in the first case.
For the same reason, if you jump off a table, it's preferable to bend your knees upon landing. The more you bend, the more comfortable the jump is going to be.
- minimizes the impact area in order to maximize pressure, and
- makes use of the body hardened spots and damping structures in order to minimize damage to self.
So, the more intense pain of a failed attempt doesn't necessarily correspond to a larger force, but it may instead simply result from the force being applied to the hand in a more damaging way. And, even if a stronger force is involved, it might fail to break the board because it's spread over a larger area or not applied abruptly enough, allowing the board to deform instead of break.
Accordingly to Newton's third law, this force is the same with that of the hand's exerted to the board in each case.
This is an incorrect assumption, and perhaps part of the problem you are facing.
As mentioned in other answers; an important factor is the time required to slow down your hand. Something softer will compress more. The more it compresses, the longer it should take to slow down the hand (assuming the same impact speed for comparison). Since it takes longer to slow down, it has less acceleration on your hand. Net force is proportional to acceleration; so the less the object deforms; the more force it applies to your hand.
Another important factor that seems to have gotten overlooked in these answers is the difference in motion between going through a block, and hitting into it.
Another way to think of applied force is to think of the acceleration required to reach your new speed.
In the case of going through the block; your final speed is not $0$. You still have some velocity after breaking it, which is used to keep your hand going.
When you hit into the block, your hand stops completely. This means the force applied by the block to your hand is based on the acceleration required to stop your hand.
When you go through, since you still have some velocity, you must have experienced less acceleration due to the block than in the scenario where you are stopped by the block.
This means that not only does the hardness of the surface effect the impact force; but the impact force is actually less when you go right through, as compared to when you hit the block and are stopped by it.
That's not really to say that it should be easier to go through a block than to hit into it; but really it means that blocks that you are able to break through will apply less force to your hand than ones that you cannot break through; even if impact times are the same.