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Totally agree with the above comments on about the difference in light, subject interaction, composition, etc making a much bigger difference than crop vs FF. This is especially true for stopped down, strobe lit shots where it can be very difficult to tell the difference in the final image.
The DOF difference has already been mentioned, full frame will give you shallower depth of field by approximately 1 1/3 aperture stops when compared to 1.6 crop.
The other byproduct is the overall "look" of the whole image, by virtue of the fact that full frame uses the whole image circle of the lens and crop uses just the central portion of the image circle of the lens. This means that FF will show more of the softness/vignetting/aberrations at the periphery of the image where crop will show just the central portion. This can be a good thing or a bad thing depending on your perspective and whether you like that overall look.
A second, and more subtle, factor is that full frame requires less enlargement from the sensor size to any given print size. A crop image has to be enlarged more to make the same print (not pixel dimensions, but actual physical dimension). As you enlarge an image more, you reduce local contrast (MTF). So the FF image has a slight advantage because it demands less resolution of the lens. This is why a print from a 4x5 or 8x10 view camera with shallow depth of field can look so sharp and contrasty at the plane of focus compared to the OOF areas, the lens is not having to produce a very high resolution and therefore the contrast (MTF) can be better.
Finally, FF vs crop forces you to shoot with different focal length lenses to get the same field of view. If you're shooting prime lenses, that also forces a comparison between the two involved lenses and which one is better.
Anyhow, enough words. Here's what the difference looks like in a practical, shallow DOF situation. Pay less attention to the DOF difference and instead look at the image as a whole to see what you like.
85L II at f/1.2 versus 50L at f/1.2, both from the same shooting position, 50L shot cropped to 1.6 field of view. No lens correction applied, same settings in LR.
50L with 1.6 crop
85L full frame