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We show that the classical (mean-field) description of far from equilibrium superconductivity is exact in the thermodynamic limit for local observables but breaks down for global quantities, such as the entanglement entropy or Loschmidt echo. We do this by solving for and comparing exact quantum and exact classical long-time dynamics of a BCS superconductor with interaction strength inversely proportional to time and evaluating local observables explicitly. We further identify a regime where nonequilibrium quantum fluctuations of anomalous averages (superconducting order) are unusually strong, such that the thermodynamic limit does not commute with the adiabatic and strong initial coupling limits for these local observables. The long-time steady state of the system is a gapless superconductor whose superfluid properties are only accessible through energy resolved measurements. This state is non-thermal but …