Clearly there should be a limit where both are one and the same, isn't it?
I think the underlying contrast you want to make is the continuum model of a solid vs the atomistic model of a solid. The elastodynamic equations come from the continuum model, and the phonon equations come from the atomistic model. That also hints at where the two agree: in the limit of wavelengths much longer than the spacing of the atoms, the material can often be treated as a continuum, and the elastodynamic and phonon equations will converge --- at least for acoustic phonons.
However, optical phonons have no continuum equivalent; they exist due to having more than one atom in a unit cell, and continuum models cannot reproduce the effect.
How is the thermal conductivity or the time-relaxation rate related to the elastic coefficients of wave propagation in linear elasticity?
There's not a direct connection for a few reasons:
A continuum model allows for an arbitrarily small wavelength whereas the atomistic model does not. If you try to calculate thermal conductivity with the continuum model, those arbitrarily small wavelength waves will result in problems --- especially at high temperatures. This is a little like the ultraviolet catastrophe for light.
The continuum model leads to linear dispersion relations ($\omega \propto k$), which is correct in the limit of small $k$ but is otherwise wrong.
The continuum model cannot explain the relaxation time because a non-linear dispersion relation is required for phonon-phonon scattering. Linear dispersion relations allow waves to "pass though" each other without interacting, but an interaction is required for phonon-phonon scattering to exist.
The continuum model cannot explain some other sources of scattering (and the relaxation times they result in) such as mass-difference impurity scattering, which arises from some atoms having different masses than the rest of the crystal (e.g. most of the sample is Si 28, but other isotopes of Si are there in small quantities). The continuum model cannot account for this because it does not allow for atoms.