Charon stays in the same place in Pluto's sky, but the moon moves across Earth's sky. What causes this difference?
Pluto and Charon are tidally locked to each other and always show the same face to each other.
When a moon orbits a planet gravity exerts tidal forces on the two bodies.
In the case of a moon the tidal forces exerted by the planet slow down the period of rotation of the moon. The angular momentum is transferred to the moon's orbit. Eventually the moon's rotation period becomes the same as its orbital period. This is called tidal locking which is a minimum energy state.
Most moons are tidally locked to their planet as they are smaller than the planet and the process happens relatively quickly.
In the case of Pluto and Charon, they are both of similar size. As a result they have become tidally locked to each other. This means that they always show the same face to each other.
In the case of the Earth and Moon, the Moon is tidally locked to the Earth and hence always shows the same face to the Earth. The Earth however is not tidally locked to the Moon. Hence the Moon moves across the sky.
The Moon is of course exerting tidal forces on the Earth which causes tides in seas and oceans. This is causing the Earth's rotation to slow down and the Moon to move away from the Earth. Eventually the Earth and Moon would be tidally locked to each other, but that will be after the sun has become a red giant and consumed them both.