What is the clear cut difference between isotropic and anisotropic spin exchanges? When are spin exchanges said to be isotropic or anisotropic?
I have read several articles on this and can not differentiate these concepts properly.
 A: It depends on context. Isotropy means "the same in all directions". In the spin context, this can sometimes mean actual spatial directions, but usually it means directions in "spin-space".
That is, an isotropic exchange interaction is one where the x-components of the two spins interact the same as the y-components, and both interact the same as the z-components. For two spin-1/2 objects, the isotropic interaction is the Heisenberg interaction (S dot S). Another possible spin-spin interaction for S=1/2 is Dzyaloshinskii-Moriya, which arises from spin-orbit coupling. This provides a mechanism for anisotropic interactions -- spin-space is connected to the (generically) anisotropic real-space.
A: When are spin exchanges said to be isotropic or anisotropic?
I agree with the other answer, that the precise meaning depends on the context. My answer concentrates on the specific context of ab initio calculations in condensed matter physics/chemistry. 
Calculations are often performed using density functional theory (DFT) with on-site Coulomb corrections, also known as  DFT +U (-J), to correct for the excessive delocalisation of valence electrons due to inexact exchange errors. On-site corrections come in a myriad of forms, but by far the two most common types are:
i) The Spherically Averaged effective on-site correction, +Ueff, by Dudarev et al.[1]
&
ii) The Anisotropic Rotationally Invariant two term correction, +U-J, by Liechtenstein et al.[2]
In the Spherically Averaged +Ueff approximation, the exchange interaction is 'isotropic', in the sense that only monopole terms in the Slater integral parameterisation of the on-site interaction are considered, so there is no angular dependence in the interaction and no off-diagonal exchange matrix elements. The on-site exchange interaction J can therefore effectively be absorbed into the on-site monopole direct Coulomb interaction U, allowing the use of a simple single effective on-site term, +Ueff (=U-J), in this isotropic approximation.
For the Anisotropic Rotationally Invariant two term correction, +U-J, the difference is that the parameterisation of the on-site interaction is a bit more realistic and includes the multipole Slater radial and angular integrals (F2, F4, c0, c2, c4, etc). This leads to a description of on-site exchange described by matrices of dimension 2(l)+1 by 2(l)+1 for each spin channel which describe the on-site exchange between electrons within different spin and orbital quantum numbers. Anisotropic exchange in this context effectively means taking into account that Coulomb exchange is different between orbitals with different shapes, for example, exchange is different between spin up x2-y2 and down 3 z2-r2 orbitals than between spin up x2-y2 and down xy, due differences in symmetry, or differences in the shapes/angles and lengths of lobes between the two cases. Anisotropic exchange formulation is important for strongly correlated Mott insulators, materials where Hund's coupling is critical to the orbital structure and systems with non-collinear magnetic ordering.
To clarify the exchange anisotropy I mention comes from the real space orbital symmetry dependence, not necessarily spin non-collinearity.
[1] - http://journals.aps.org/prb/abstract/10.1103/PhysRevB.57.1505
[2] - http://journals.aps.org/prb/abstract/10.1103/PhysRevB.52.R5467
