The structure of the nucleon is studied by means of deep-inelastic neutrino-nucleon scattering at high energies through the weak neutral current. The neutrino-nucleon scattering events were observed in a 340-metric-ton fine-grained calorimeter exposed to a narrow-band (dichromatic) neutrino beam at Fermilab. The data sample after analysis cuts consists of 9200 charged-current and 3000 neutral-current neutrino and antineutrino events. The neutral-current valence and sea nucleon structure functions are extracted from the x distribution reconstructed from the measured angle and energy of the recoil-hadron shower and the incident narrow-band neutrino-beam energy. They are compared to those extracted from charged-current events analyzed as neutral-current events. It is shown that the nucleon structure is independent of the type of neutrino interaction, which confirms an important aspect of the standard model. The data are also used to determine the value of sin2θW=0.238±0.013±0.015±0.010 for a single-parameter fit, where the first error is from statistical sources, the second from experimental systematic errors, and the third from estimated theoretical errors.