Here's a more physical argument for why damage is caused by kinetic energy instead of momentum. When I say "caused by" I mean that two projectiles with the same kinetic energy will cause the same amount of damage, whereas two projectiles with the same momentum may not.
When a projectile (a baseball, for example) impacts on an object (a glass window), there can be two results: the object stops or reflects the projectile with no damage, or the projectile penetrates the object, causing damage until it comes to rest inside the object or breaks through the other side. By Newton's Third Law, the force that the window exerts to stop the baseball is equal to the force the baseball exerts on the window. So, even if the window doesn't break, it feels a force and bends in response--the greater the force, the greater the bending. A window can only bend so far before it breaks, so there is a maximum force that can be applied to the window before it breaks. By Newton's Third Law again, there is a maximum force that the window can exert on the ball.
So, as the ball is thrown harder against the window, it takes greater and greater force to stop the ball, causing greater and greater bending in the window. Once the force is great enough, the window breaks and the ball continues on its way at a slower speed due to the force of the window.
Now imagine, instead of a single window pane, there is a large stack of window panes in the way of the baseball. The baseball will break through a certain number of windows until it slows down enough for one window deep in the stack to stop it. What does each window do? When the ball makes contact, the window bends in response until it reaches it's breaking point, applying a force to the ball over a certain distance. This means that each broken window does work on the ball, which means the ball does some amount of work (force over a distance) to break the window. If the windows are identical copies, each will take the same amount of work to break, and the amount of damage is the number of windows broken, which is the amount of work done on the windows, which must be equal to the kinetic energy of the ball.
Momentum doesn't matter because it doesn't matter how much time a ball pushes against a window. You can rest a ball on top of a horizontal pane of glass and nothing will happen, even as the ball applies an unlimited amount of impulse (force times time) to the window. What matters is the distance that the ball deforms the window, which requires a certain amount of force, and exerting a force over a distance is work, i.e., energy.
You can try an experiment for yourself. Get a box of sand and some hunks of clay. You can measure damage by measuring the size of the crater formed when the lumps of clay are dropped onto the sand. You'll need to perform at least 4 drops: 2 where different masses of clay impact with the same kinetic energy and different momenta, and 2 where different masses impact with the same momentum but different kinetic energies. One of these pairs of drops should make similarly sized holes and the other pair should make different sized holes.