However, the voltage is small between any two nearby points on the wire itself. In our gravitational analogy, it is like being on top of a cliff. With the gravitational case, falling a few feet won't hurt even if you are at the top of a mountain, but falling to the ground from the mountaintop would. The voltage across the bird is minimal. In addition, the current flows through the bird's legs rather than through its heart or brain.
Some organs are more sensitive to the electrical interference, or more critical when interrupted.
Their feathers may get puffed up because they will suddenly gain some charge when they touch the wire. The third rail is the high-voltage rail.