Electronic bands alone do not determine whether a solid conducts. Conductivity depends crucially on which states are occupied and whether nearby unoccupied states are available for electrons to respond to an electric field.

At zero temperature, electrons fill available states up to the Fermi energy. A partially filled band contains both occupied and nearby empty states, so electrons can change crystal momentum under weak fields. This is the basic band-theory picture of a metal.

If the highest occupied band is completely filled and separated from the next empty band by an energy gap $E_g$, weak fields cannot rearrange electrons within that full band into nearby empty states. The solid is insulating at zero temperature.

A semiconductor has the same qualitative filled-band/gap structure as an insulator but with a sufficiently modest gap that thermal excitation, light or doping can produce mobile carriers. Exciting an electron from the valence band to the conduction band leaves an empty state—a hole—that behaves as a mobile positive carrier in the band description.

The distinction is therefore not simply 'metals have bands and insulators have gaps'. All crystals have bands. The important questions are where the Fermi level lies, whether the relevant band is full, and how large the gap is compared with available excitation energies.

This band-filling viewpoint connects Bloch states and Fermi statistics to the semiconductor carrier models used in electronics.