In turbulent superfluid He II, the quantized vortex lines interact via the classical Biot-Savart law to form a complicated vortex tangle. We show that vortex tangles with the same vortex line density will have different energy spectra, depending on the normal fluid which feeds energy into the superfluid component, and identify the spectral signature of two forms of superfluid turbulence: Kolmogorov tangles and Vinen tangles. By decomposing the superfluid velocity field into local and nonlocal contributions, we find that in Vinen tangles the motion of vortex lines depends mainly on the local curvature, whereas in Kolmogorov tangles the long-range vortex interaction is dominant and leads to the formation of clustering of lines, in analogy to the 'worms' of ordinary turbulence.