The two can be distinguished by the anisotropy of the anomalous part of the conductivity tensor in the case of Weyl metals; assuming the z-direction to coincide with the direction of the applied magnetic field, in the case of Weyl metals only δσzz is non-vanishing. For details, consult the paper by TD Son, and BZ Spivak (Phys. Rev. B 88, 104412 (2013)). See in particular Eq. (17) herein.

Further, the magnetic-field dependence of the conductivity tensor is also different in the two cases. For the Weyl metals, δσzz scales like the square of the amplitude of the applied magnetic field (at least in the region of weak-fields), while in other cases the field dependence can be different; depending on the values of various parameters, the size of the correction to the conductivity scales like the logarithm of the amplitude of the magnetic field in two dimensions, and like the square root of this amplitude in three dimensions. For details, consult the paper by A Kawataba (Solid State Commun. 34, 431 (1980)), as well as Chapter 1, Electron-Electron Interaction In Disordered Conductors, by BL Altshuler, and AG Aronov, of the book Electron-Electron Interactions in Disordered Systems, edited by AL Efros, and M Pollak (Elsevier, Amsterdam, 1985). See in particular Eq. (2.11), p. 11, herein.

Ps. When considering the transport properties of systems, in particular the conductivity tensor in its dependence on the applied magnetic field, one must be mindful that these properties crucially depend on the electronic structure of the underlying system. For instance, it is well-known that semi-classically the dependence of the conductivity tensor on the applied magnetic field crucially depends on whether the semi-classical orbits are all closed or whether there are some (at least one) which are open. For details, consult Chapters 12 and 13 of the book Solid State Physics, by Ashcroft and Mermin.

     Dear Utpal,

    As is well known the breaking of the chiral symmetry coupled to Weyl spinors (i.e. Dirac spinors with zero mass) is associated to a non-conservation of the fermion current (electrons or holes in solid state physics). The non conservation is proportional to a Chern number which measures the non topological triviality of the band structure in the first Brillouin zone.

   The first people to treat this issue quite seriously was in the paper, where you can see how to distinguish this current in a solid state system:

     H.B.Nielsen, Masao Ninomiya, Phys.Rev.B 130, 389 (1983)

and more related with your question you have a paper:

     Cheng Zhang et al., Detection of Chiral Anomaly and valley transport in Dirac       semimetals, ArXiv 1504.07698 (2015) 

  Sorry, the first reference I have confused the review

  H.B.Nielsen, Masao Ninomiya, Physics Letters B 130, 389 (1983)

Respected Behnam sir and Daniel Sir,

     Thank you very much for your valuable references.