8 bits = 256 values for each colour
16 bits = 65535 values for each colour
What this basically translates to is that a 16 bit image will have a much larger dynamic range and you will have less clipping, especially in the highlight areas of an image. A quick way to see what I'm talking about is to save the same image as an 8 bit and a 16 bit image. Do a curves adjustment exactly the same on both. Then go check out the histogram.
Keep in mind though that Raw files are essentially 12 bit (12 bit having 4096 values). Since programmes offer editing either in 8 bit or 16 bit mode it is better to use 16 bit. 16 bit mode being able to accommodate 12 bit images makes it a better option.If you use 8 bit then you are essentially compressing 4096 values into 256 and thus you are downsampling and losing information, not to mention that you image can have clipping or blown out highlights.
Jpeg = 8 bit
Raw & Tiff = 16 bit
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