Difference between coherent scattering and elastic scattering

This is related to a fundamental misunderstanding of coherence (which is very common). Coherent processes are typically associated with wave coherence, so coherent scattering means scattering of two waves. This is in opposition with scattering between a wave and something that is not a wave (or, by some interpretations, something that does not behave like a wave, like a macroscopic bunch of matter such as a bouncing ball).

First need to understand that when comparing Coherent scattering and elastic scattering refers to a wave that has been absorbed follow by a re-emission at the same frequency of the incident wave. The coherent means that the re-emission of the wave maintains the same direction of the incident ray (like laser beams). Elastic scattering is any other new re-emission but traveling in a different direction.

In the literature, the two terms are often mixed up and used differently. In my answer, I will use the most distinct and most common interpretation of the two.

In simple terms, elastic scattering is about energy. Specifically, "the kinetic energy of the scattering particle is conserved in the center-of-mass frame" (see wikipedia). Inelastic scattering is then a process where the scattering particle loses energy (see wikipedia).

Coherent vs. incoherent scattering is not about energy, it is about the phase or fluctuations of a the wave or scattering particle. There does not seem to be a wikipedia article on coherent scattering, but there is one for incoherent scattering. Coherent scattering is then the case where the scattered particle or wave has a fixed phase relation relative to the initial wave, such that you can observe interference between the two. This coherence can be destroyed by fluctutations of the scattering medium or by quantum effects such as inversion of level systems.

Importantly, all combinations of the two categories are allowed. That is coherent elastic scattering, incoherent elastic scattering, coherent inelastic scattering and incoherent inelastic scattering are all different physical processes.

In mathematical terms, the processes can be categorized by properties of the difference between the initial and final state of the scattering particle in the scattering process $psi_i rightarrow psi_f$. For elastic scattering, $psi_i$ and $psi_f$ have the same energy, for inelastic scattering they do not. To investigate incoherent scattering, one usually leaves the framework of wavefunctions and introduces density matrices or statistical ensembles. A common example of coherent and incoherent scattering is the famous Mollow triplet in resonance fluorescence, where both processes occur simultaneously.