In regard to the "nail example", or any other example involving bending,breaking, or penetration for that matter- will require a certain minimum pressure to occur. This is because strain i.e. deformation(from applied force) per unit original dimension is proportional to stress(=force per unit area) and not force. This nature of deformation --strain's proportionality to stress (to first order) is why pressure is a useful quantity (in fact this is how moduli of elsticity are defined).
For the same amount of force the stress produced on a surface by the nail's tip would be much larger than that by its head- hence more strain i.e. higher penetration.
Stress is another name for pressure in this context. Pressure is defined as the magnitude of force normal to an area per unit area. From this definition
some key differences are clear:
Pressure is a scalar--its always defined normally to a surface--a tangential non-zero force produces zero pressure.
Pressure is not completely defined by force alone--the area element on which the force acts is also needed-a small force in a small area produces no less a pressure than a larger force in a larger area
What about pressure in fluids? In a(n) (isotropic)fluid any arbitrary point experiences a pressure that is same from all directions--in another words if a differential area dA was to be placed at that point, then regardless of its direction it would experience the same force.
What about gasses? There the pressure is defined by finding the average force imparted to the walls of a container per unit area--the same definition just the computation is different.( given the velocities with which the gas particles hit the container have a distribution and so each individual hit doesn't produce the same pressure.)
Pressure as in atmospheric pressure? Its just the weight of the mass of an infinite(in theory) air column above sea --again--per unit area.
There are countless examples where pressure--and not just force--plays the deciding role not the least important of which is the threshold in safety release valves or the nature of sound.