How large an energy difference in Hartree can be used to distinguish two different structures?
I wouldn't use energy as a general criteria to discriminate conformations; a root mean squared deviation (RMSD) of superimposed structures would be a better metric. This is because two very different conformations may have very similar energy. For example, if you perform a relaxed energy scan of one of the dihedrals of cyclopentane, you get nearly the same energy for most of the dihedral angles, bu very different ring conformations.

If you found an energy difference of 0.0000001 a.u. and the superimposition of the conformations is nearly perfect, I'd say they are in the same minimum of the energy surface.