The answer depends on quality of the source file and the intended use of the image. When you work with well exposed photograph and don't edit it intensively, the difference between 8bit and 16bit isn't significant. However in case you need best possible quality there is a difference.
Even if you save the final result in 8-bit file (e.g. JPG only supports 8-bit mode), there is a difference depending how you edit your file.
Whether it's a big deal or not is rather subjective.... but the difference is pretty obvious.
I made a simple example - took a well exposed image, didn't apply very much editing, didn't have too many layers. There are 2 frames in the animated gif: one is the result of 16 bit editing, converted to 8 bit at the end; the other frame is result of 8-bit editing - i.e. same file was first converted to 8 bit, then editing applied. It enlarged to 400% to make the difference more visible.
In case you have underexposed image, in case you have higher ISO, in case you apply intensive editing - you will see much bigger difference between 8 and 16 bit. But even this example above is sufficient for me to do my editing in 16 bit.
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