What is the difference between vertical turbulent entrainment and turbulent diffusion in the ocean?
I would say that different authors might want to distinguish deep convective mixing with entrainment from purely diffusive or gravity wave-driven mixing. Deep mixing occurs mainly within certain zones (around Antarctica, and in the Labrador sea) because it requires a very strong cooling of the surface to trigger convection. Diffusive or gravity wave-breaking mixing can occur anywhere. Diffusion could include direct processes (like double-diffusion or "salt fingering") or indirect processes such as cabbeling or thermobaricity. If their budgets are based on models, these classes of processes are likely output separately anyway, so it's an easy division to make.

I would break it down as follows, with an eye toward model output:


Vertical advection comes from the advection by resolved or implied vertical velocity. This would include Ekman pumping and any other vertical velocity you might get from an omega-equation.
Entrainment-detrainment comes from convective processes. These require special handling, since many ocean models until recently would not resolve these processes. (They might occur in areas that are only a few kilometers in width, far below the lateral resolution.) These processes can mix together entire parts of the water column at once and often do not care about vertical gradients; they are "non-local".
Diffusive processes are very local/small-scale and depend on vertical gradients of heat and salt. Cabbeling, thermobaricity, and double diffusion are some examples. I include gravity wave-driven mixing here because although the gravity wave generation process is non-local, when they break they can only mix locally within the water column.


This kind of breakdown makes sense for a global ocean model, where these three things are handled by three separate modules. For super-hi-res regional models and non-hydrostatic models, advection and convection start to merge together, and if the model is truly eddy-resolving, entrainment becomes a mix of advection followed by diffusion. How you budget these processes depends on the spatial and temporal scales of the model.