One major difference to keep in mind: the LUTs are designed and work in the display-referred part of the pipeline - after filmic/sigmoid.
Another thing: LUTs can be large and much more precise in terms of specific color tweaks than profiles and can even be totally non-linear. That’s how a Look Up Table works. It takes an input value and just delivers the corresponding output value. This makes them a fast one-stop-shop because they can contain multiple “adjustments” all in one.
They can have color shifts, inversions and all kinds of funny effects you can not replicate with parametric tools like color calibration. But for their full potential they also need to work with a clearly defined basis or their transformations can have strange side-effects.