Nobody seems to be thinking scientifically.
Why do helmets protect you?
1) They dissipate the energy of the initial impact over a larger surface area due to the compression at impact.
2) The compression of the helmet slows the acceleration of the head less rapidly than pavement does.
Let's do some physics.
The first aspect has to do with pressure, or Force over an area. P = F/A. F being the Normal Force.
Think about it like this, A man weighing 70 kg (~154 lbs, Normal Force = 700 Newtons) steps on your foot and a woman weighing 45 kg (~99 lbs, Normal Force = 450 Newtons) steps on your foot. Now, you would think the man stepping on you would hurt more, that is true, unless pressure is distributed differently. Let's assume the heel of the man's shoe is 7 cm x 5 cm (Total surface area of 35 cm, or .35 meters) he woman is wearing high heels and her heel is a 2 cm circle (Surface area of about 6 cm, or .06 meters)
Now let's do the calculations:
For the man: P = F/A, P= 700/.35, P= 2000 Pascals
For the woman: P = F/A, P= 450/.06, P= 7500 Pascals
Due to the smaller surface area, despite the lower weight, the woman stepping on our foot would hurt more.
The same is true if you fall with a helmet vs. without a helmet. The helmet compresses and creates a larger surface area of impact to the pavement than your skull. Your skull may compress a bit, but in doing so the bone is weakened and will likely result in a fracture.
Let's do some similar calculations simulating a bicycle crash from a standstill.
Assume: Your head weighs 5 kg, Helmet compresses to a surface area of impact of 27 cm^2 (.27 m^2), Skull compresses to 9 cm^2 or .09 m^2.
Force of head at point of impact: F=MA, F=5*9.81, F=49.05
Pressure- Helmet: P=F/A, P= 49.05/.27, P=181.6... KPascals
Pressure- No Helmet: P=F/A, P=49.05/.09, P=545 KPascals
That is a significant difference.
This does not take into account the factor that is most significant in causing lasting brain damage. That is, the rapid deceleration of the skull. The brain is suspended in liquid inside the skull, this provides a cushion against trauma to the head. However, if the skull is decelerated very quickly, as it is on impact with pavement, the brain strikes the inside of the skull causing brain damage. A helmet helps to slow the deceleration of the skull the same way an airbag slows the deceleration of the body in a car crash. A helmet will get crushed in a direct impact, in doing so, the skull is slowed to a stop over a longer period of time than if there were no helmet.