= Solution
The <tight-binding approximation> starts with localized atomic orbitals. Their overlap allows hopping, broadening each atomic <energy> into a band. For example, a single orbital on a simple cubic lattice with nearest-neighbor hopping gives $E(\boldsymbol k)=E_0-2t[\cos(k_xa)+\cos(k_ya)+\cos(k_za)]$. Small overlap means a narrow band; orbital symmetry and hopping signs determine its extrema and dispersion.
The <nearly-free electron model> starts instead with free-electron parabolas, folded into the <Brillouin zone>, and treats the periodic crystal potential as a perturbation. Near a Bragg plane, states differing by a reciprocal vector become degenerate and mix, opening a gap approximately $2|V_{\boldsymbol G}|$. Away from such degeneracies, the <energy> remains close to the free-electron dispersion. These are complementary limits of the same Bloch-band framework, not different counts of electronic states. Both can give metals or insulators depending on filling and band overlap, but their natural basis functions and useful expansion parameters differ.
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