Q.In terms of band theory, what is the difference

$(i)$ In a conductor the valence and conduction bands overlap (or the valence band is partly filled), so electrons move freely; in an insulator a large forbidden energy gap separates the filled valence band from the empty conduction band, so electrons cannot be promoted. …

Band theory basics.

Electrons in a solid occupy energy bands: a filled valence band and a higher conduction band, separated by a forbidden energy gap (band gap).

(i) Conductor vs insulator.

• Conductor (metal): the valence and conduction bands overlap, or the valence band is only partially filled. Electrons move into empty levels with negligible energy, so it conducts freely.
• Insulator: a large band gap separates a completely filled valence band from an empty conduction band. Electrons cannot gain enough energy to cross, so no conduction.

(ii) Conductor vs semiconductor.

• Conductor: essentially no band gap (overlapping bands) — high conductivity at all temperatures. …

Frame all three by the forbidden energy gap between valence and conduction bands: overlap/no gap (conductor), large gap (insulator), …

• Describing conduction without mentioning the band gap at all.
• Saying an insulator has a small gap (it is large; a small gap is a semiconductor). …