I think there's a fair distinction between them, both in appearance and usage.
An armillary sphere is a model of objects in the sky (on the celestial sphere), consisting of a spherical framework of rings, centered on Earth or the Sun, that represent lines of celestial longitude and latitude and other astronomically important features, such as the ecliptic. As such, it differs from a celestial globe, which is a smooth sphere whose principal purpose is to map the constellations.
In layman terms, while an armillary sphere (eg. post #4217417) is an object with a bunch of moving parts and may or may not have a center sphere, a celestial sphere is a map of the night sky (eg. post #5262224).
Gardened both tags, removed mistags.