In newtonian mechanics gravity is a so called body force, that acts throughout the volume of a body. Since a body force produces the same acceleration for every small mass element of a body, it doesn't create internal mechanical stress between parts of the body.
(Well, for extended bodies gravity can create tidal forces, but they are very small for everyday objects. Tidal forces only exist in an inhomogeneous gravitational field.)
In contrast to that, the normal force from the ground that holds you up against gravity is a surface force, that acts across an internal or external surface element in a material body. Surface forces create internal mechanical stress inside a body. While body forces can directly act on any mass element inside a body, surface forces have to "travel" from one mass element to the next.
For example when standing on the ground the normal force pushes against you feet, your feet push against your leg, you leg against your hip and so on. You can feel a difference when the surface force acts on a different part of your body. While standing you feel pressure on your feet, but while sitting you feel the pressure on your buttocks.
The deformation of the corresponding sensory cells due to mechanical stress gives the perception of a force. Therefore you don't percieve any force, when only gravity is acting on you while in free fall.