both TL snd UL approaches play a role in large deformation analysis only (there is no difference between them for small deformations and small strains). I don't really know the procedure used in Abaqus, but several FE codes prefer a TL approach for hyperelasticity resp. if the constitutive equations are functions of the deformation history as a whole. UL approach is used e.g. for rate constitutive equations, including the case of hypoelasto-plasticity where the "elastic" part is not derived from a potential (also referred to as the "ad hoc" extension of small strain elasto-plasticity to large deformations). Hope this helps.
Ansys uses TL for hyperelasticity. If you have dissipative effects (more generally where the constitutive model requires history dependent variables), as in plasticity, creep etc UL is normally used.
I have an additonal question. Programming hyperelasticity in Ansys or Abaqus, there exist two ways:
UserMat
UserHyper
In the case that the usermat and the tangent is correctly programmed in terms of the Jaumann rate. Why is the convergence behavior of usermat not as good as for userhyper? What are the differences between UserMat and UserHyper? The documentation is rare on this topic. Is Userhyper utilizing a TL formulation and UserMat a UL?