Fully developed homogeneous isotropic turbulent fields, computed by direct numerical simulation, are compared to divergence-free random fields having the same energy spectrum and either the same helicity spectrum as that of the turbulent data, or vanishing helicity. We show that the scale-dependent velocity flatness quantifies the spatial variability of the energy spectrum. The flatness exhibits a substantial increase at small scales for the turbulent field, but remains constant for the random fields. A diagnostic, the scale-dependent helicity, is proposed to quantify the geometrical statistics of the flow, which shows that only the turbulent flow is intermittent. Finally, statistical scale-dependent analyses of both Eulerian and Lagrangian accelerations confirm the inherently different dynamics of turbulent and random flows.