The fact is that, in classical Lagrangian field theory on a fibre bundle Y->X, an energy-momentum current, by definition, is a symmetry current along a vector field u on Y, which has a certain non-zero projection s on X. Such a lift u of s is not unique, and therefore an energy-momentum current fails to be unique. The difference of two different lifts u and u' is a vertical vector field v=u-u' on Y possessing a zero projection on X Symmetry currents along vertical vector fields are Noether currents of internal symmetries of a Lagrangian. Therefore, different energy-momentum currents differ from each other in Noether symmetry currents. Moreover, any conserved energy-momentum current in general contains a Noether component, determined by internal symmetries of a Lagrangian.
G.Sardanashvily, Energy-momentum conservation laws in gauge theory with broken gauge aymmetries, arXiv: hep-th/0203275
G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Fiel Theory (WS, 2009)