What kind of ICC profile, and what kind of LUT?
ICC profiles can represent many things.
And there are many different things called LUTs.
Assuming you mean a 3DLUT, it's not generally possible -- because ICC profiles are a superset of what can be expressed in a 3DLUT. (but 3DLUTs can be converted to devicelink profiles easily enough)
Some RGB devicelink profiles, and some abstract profiles can be converted to 3DLUTs if you specify the exact conditions for the 3DLUT (which encoding, what primaries, what whitepoint, etc.).
Again, a "plain conversion" isn't always possible.
It depends on what kind of profile you are starting with, and what kind of LUT you are trying to create.
And even when it is possible, you need more information -- ICC profiles are standardized, and tied to color appearance standards. 3DLUTs are not standardized, and contain pretty much random mappings (so you'll get different results for different encodings and colorspaces).
ICC profiles generally are tied to CIE L*a*b* or CIE XYZ, while 3DLUTs can be designed for linear, log, Rec709 encodings and many different primaries and whitepoints.
It's like you're trying to use a current smartphone (ICC profiles) on a network from 1990 (LUTs) - in some cases it might work, but only in limited circumstances, and you certainly can't use the full capabilities of the modern tech. To make it work at all you'd need details on the exact interchange needed.
Another way of explaining it: 3DLUTs are more comparable to adjustment layers (or groups of adjustment layers) than ICC profiles. Only two types of ICC profiles (device link and abstract) are even comparable to 3DLUTs (and still not quite directly). Display profiles, print profiles, and colorspace profiles really don't match the 3DLUT concept.
Display profiles are only "applied" when your document data is being sent to the display.
Photoshop is a compositing program.
But some film compositing applicaitons are many years behind on color management.
When a color managed application sends your document data to the display, it first converts the document color values into the values needed by the display to correctly show the colors in your document (preserving the appearance). This uses the document profile and the display profile to do the conversion from the document colorspace to the display colorspace. This way you see the best representation of your document, regardless of the type of display being used, and can have some confidence that you are seeing your document correctly. The document data is not harmed, but only converted as needed for display, and you can use the document on multiple displays and the colors will be converted correctly for each display.
Non color managed applications tend to copy the document data directly to the display (some try to apply a 1DLUT to account for the value encoding, but that rarely works well), regardless of the document colorspace or the display colorspace (keeping values, mangling the appearance). That gives you color that cannot be trusted - because the document values rarely match the actual display. Unfortunately, this is still the current state of many film compositing applications, and more than a few video applications. With this approach you will see something different on each display.
As for Photoshop being a compositing application - that should be obvious.
Photoshop is an application widely used for compositing still images and video/animation.
Beaver, maybe you could describe what you're trying to accomplish.
Your questions imply just enough knowledge to begin to be dangerous. It would help others teach you if you'd just describe what it is you're trying to do. What you've asked so far is the equivalent of "what carburetor is better?"
Without having stated what software you're working with or what your goals are, no one can answer the question any better than Chris already has, above.
If you're looking for a free custom-tailored education on how LUTs work, maybe Googling for more info with which to educate yourself could be helpful?