The following Marcy image shows the difference between a 1D LUT and a 3D Cube.
The initial image is a 1D LUT and when you hover your mouse over it the image will show the result of a 3D Cube, both LUTs being based on exactly the same data.

As can be seen the major differences are in the levels of saturation as a 1D LUT cannot alter saturation separately to brightness.
This doesn't mean 1D LUTs are of no use, as they can be used to great effect a lot of the time where 3D LUTs are really overkill. On-set viewing where the actual viewing conditions are not good is a perfect application for example, especially as the various cameras that shoot in an 'extended range' mode only adjust their effective gamma, so a 1D LUT can be perfect in displaying a 'viewing' image.
Next Page - How to use LUTs
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