Human fingertips and skin in general are extremely precise high resolution pressure and vibration sensors that robots almost completely lack. Tactile feedback takes up the second most human sensor bandwidth behind vision. Imagine doing anything without it- interacting with the world through a puppet.

The joints in a human arm are direct-drive series elastic actuators with pretty incredible sensitivity. If you were holding a big heavy box and I bounced a ping pong ball off it, you'd probably feel the vibration. That's kind of a big deal. You may not be able to measure absolute weight well, or even relative weight, but you can detect quick changes with amazing precision. Lifting any heavy or large object is inherently wobbly even for a robot, and that sensitivity is important.