1. Identify the forms of the elements in their standard state

   • Oxygen exists as $\mathrm{O_2}$ molecules.
   • Sulphur exists as $\mathrm{S_8}$ molecules.

2. Compare intermolecular forces

   Both oxygen and sulphur are nonpolar molecular substances, so the main intermolecular force between their molecules is London dispersion force.

   • $\mathrm{O_2}$ is a small diatomic molecule.
   • $\mathrm{S_8}$ is a much larger polyatomic molecule.

   Larger molecules have stronger van der Waals forces, so much more energy is needed to melt or boil sulphur than oxygen.

3. Relate this to the given options

   The key reason for this difference is the difference in molecular nature / atomicity:

   • Oxygen: atomicity $=2$
   • Sulphur: atomicity $=8$

   Because sulphur has much higher atomicity, its molecules are larger and experience stronger intermolecular attractions.

4. Check the other options

   • B: Electron gain enthalpy — this affects tendency to gain electrons, not melting/boiling point directly.
   • C: Atomic size — though sulphur atoms are larger than oxygen atoms, the major explanation for the large difference in physical properties here is the molecular size due to different atomicity.
   • D: Electronegativity — not the governing factor for these melting/boiling point differences.

5. Conclusion

   Therefore, the large difference between the melting and boiling points of oxygen and sulphur is best explained on the basis of atomicity.

$\boxed{\text{A: Atomicity}}$