John Rennie's answer is right from the viewpoint of General Relativity -- but since the question is tagged with Newtonian mechanics, it deserves a Newtonian answer too.

In the Newtonian framework, I think the best answer to "why don't we experience this force" is that we can't feel forces that applies to our body at all. What we actually experience with our senses are only forces between different parts of our body.

When you stand on the surface of Earth, you feel neither the gravitational pull of the Sun nor the gravitational pull of the Earth. Strictly speaking you don't even feel the contact force between your footsoles and the ground -- but you do feel the compressive force between the skin of your feet and the bones inside the foot. And to a lesser extent you feel your bones being compressed, and your flesh being stretched by hanging from your skeleton. All these internal forces balance out the gravitational pull on your body, such that there's zero net force applying to each part of it (ignoring the pull from the sun and moon), and you stay in place compared to the earth.

This is what produces the sensation of being pulled towards the earth: the internal forces in your body that resist that pull.

However, for the pull from the sun, there's nothing that balances it out. Every particle in your body simply falls towards the sun, with the acceleration given by the strength of the sun's gravitational field -- all every particle in the earth and in the air around you is doing the same, so no internal forces are needed anywhere to keep the various parts of your body in the same relative position. Therefore there is nothing to feel.

John Rennie's answer is right from the viewpoint of General Relativity -- but since the question is tagged with Newtonian mechanics, it deserves a Newtonian answer too.

In the Newtonian framework, I think the best answer to "why don't we experience this force" is that we can't feel forces that apply to our body at all. What we actually experience with our senses are only forces between different parts of our body.

When you stand on the surface of Earth, you feel neither the gravitational pull of the Sun nor the gravitational pull of the Earth. Strictly speaking you don't even feel the contact force between your footsoles and the ground -- but you do feel the compressive force between the skin of your feet and the bones inside the foot. And to a lesser extent you feel your bones being compressed, and your flesh being stretched by hanging from your skeleton. All these internal forces balance out the gravitational pull on your body, such that there's zero net force applying to each part of it (ignoring the pull from the sun and moon), and you stay in place compared to the earth.

This is what produces the sensation of being pulled towards the earth: the internal forces in your body that resist that pull.

However, for the pull from the sun, there's nothing that balances it out. Every particle in your body simply falls towards the sun, with the acceleration given by the strength of the sun's gravitational field -- and every particle in the earth and in the air around you is doing the same, so no internal forces are needed anywhere to keep the various parts of your body in the same relative position. Therefore there is nothing to feel.