What is the relationship between the celestial sphere and the local sky?

Answer 1

In this context the celestial sphere could be considered the background of stars, planets, comets, etc., such as is visible on a clear night, as distinct from the local sky would include Earth's atmosphere and weather effects such as an observer is largely limited to during the day (with the exception of the closest star, our Sun). It's safe to say these are relatively independent - objects in the sky such as meteorological phenomena, artificial satellites, etc., would move and act independent of the celestial sphere. Remembering that Earth is a planet in space moving in an orbit (revolving around its primary) and also rotating on its axis - because of which, the celestial sphere would seem to spin over the course of 24 hours, and also change throughout the year with respect to objects inside the solar system because of its position in orbit.

Answer 2

The celestial sphere is an imaginary sphere surrounding Earth onto which stars and celestial objects appear fixed, regardless of location on Earth. The local sky is the portion of the celestial sphere visible from a specific location on Earth. The local sky changes as one moves on Earth, giving rise to different celestial objects being visible at different times.