3DLUT files are used commonly in the film and video industry. They are RGB only, and lack any calibration (you will get different results with different colorspaces because they only map values, with no reference to color appearance). They generally use text formats to represent the data, and have lots of interoperability problems.
Devicelink profiles can handle RGB or CMYK data. In RGB they are similar to 3DLUTs, but represent a superset of the capabilities of 3DLUTs, and result in smaller files than most 3DLUT files because the data is binary.
CMYK devicelink profiles are used often in prepress, and are similar to 3DLUTs, but have 4 dimensions instead of 3 (which also makes the files much bigger)
Abstract profiles are the most general form -- they can take any colorspace for input and output to any colorspace, and are fully calibrated (they reference color appearance in CIEL*a*b*, not just values). These are also binary data, and smaller than most 3DLUTs. But not many programs can work with abstract profiles.
The abstract and devicelink profiles are part of the ICC profile standard, which is an ISO standard - so those formats are unambiguously defined, unlike 3DLUT files (which get interpreted differently in different software packages).
I appreciate your technical answer.
The reason I was asking was to try to make some sense as to when and why a Color Lookup adjustment layer might be used in Photoshop. I'm still not sure about how best to use it unless it's one of the Photoshop things that people say "just fool around with it." Is that the case?
Video and film editors use the color lookup adjustment (or layer) to import their adjustments used in other applications -- film stock/camera correction, artistic look, and film/projector output simulation.
Other people use it as a convenient way to capture or apply complex looks (like some of the examples that ship with Photoshop), much like some people over-use Instagram looks.
Thanks, Chris, for your patient explanations.