Why do we feel gravity on a plane? How come a plane flying at constant velocity experiences gravity?  If you were in a space capsule flying (not accelerating) you would feel weightless until you hit the ground.  Why not a plane?
This is the difference between flying and in orbit. In orbit, you are indeed falling toward the earth, but the spacecraft is too, and you're going fast enough that you keep missing the earth.

In an aircraft, because it's staying aloft due to lift, it is not falling. This is why you experience the pull of gravity on a plane.

Some aircraft are designed to feel weightlessness, see Vomit comet.