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When printed at 240 ppi, corresponding to a 30 x 20 inch image size, I can tell the two apart viewing at an arms length distance. No need to label them to get it right.
When printed at 360 ppi it becoms difficult to tell them apart, but it can be done by close inspection.
I'm not quite certain I'm following you here, so let me see if I'm getting it. You made two photographs of the same scene using a D800 and a 58mm lens. Presumably you did not change focus between them. One was shot at f/5.6 and the other at f/16.
You have provided two sample images, one from each full-sized image. Both samples were sharpened, but not necessarily quite the same way, since your goal was to choose settings that would optimize each one. You also applied some distortion correction to the f/16 example - but not to the f/5.6 example, right? I'll assume that you controlled for the three-stops slower shutter speed of the f/16 sample by using a tripod, live view (or whatever the Nikon equivalent is called), and a remote release - and that you used the same ISO for both samples.
When we look at the two 100% crops it can be seen that the f/16 sample is not as sharp as the f/5.6 example. Let's look at that first, and then go on to the observations about the print samples that we have not seen.
The pixel dimensions of the D800 are 7360 x 4912. At the moment I'm looking at two monitors. One provides 90 ppi resolution and the other 114 ppi, and they probably are fairly typical of what many people are using. One is a laptop monitor and the other is that of an attached external monitor. On my 90 ppi monitor, the test images are displayed at a dimension equivalent to cutting them out of a 82" wide monitor full image or a print that is nearly seven feet wide. On the 114 ppi monitor I am seeing an image that would correspond to its size if clipped out of a 64.5 inch print, or one that is about 5 1/2 feet wide.
So, what the 100% crops tell us is that we would see this much difference (accounting for the differences between print and screen display, of course) if we were to compare prints that are between about 65" and 82" wide - e.g. really huge! Building on that idea - and having inspected some really large (and highly regarded) prints very closely - what I'm seeing here is an indication of some very remarkable performance from your lens at f/5.6! I can't recall ever viewing a gallery print of that size that closely and seeing resolution quite this high. On the other hand, the resolution at f/16, while visibly a bit less sharp is still also quite good for an image that would be equivalent to a slice of such a gigantic print.
So, yes, you are getting somewhat more resolution at f/5.6 than at f/16 when you inspect quite closely, just as expected. I can certainly see it, too.
So, you then took the additional step I wrote about and made prints at a considerably smaller size, equivalent to a 30" long dimension. You observed that you can see a difference between the two images, and said, "I can tell the two apart viewing at an arms length distance" when viewing prints made with a resolution of 240. That resolution should be plenty high to not reveal pixelation in the print, so it seems like a reasonable choice for your print test. (I assume here that you are still talking about the f/5.6 and f/16 examples as described above.) I'm a bit surprised at what you saw, but I can't argue that you did not see what you saw, can I? ;-)
(An interesting additional test would be to show one of the prints to a viewer, take it away, show the other print, take it away, and then show one of the previous two and ask the viewer which one it matched. Or hang one at each end of a large room and allow people to walk back and forth between then and note differences. But I digress...)
The point of your observation is to support the idea that there are significant enough differences between the f/5.6 test and the the f/16 test that you can see them in a 30" wide print.
Then you went on to say that you repeated the test with higher resolution prints, or at least I think that is what you are saying: "When printed at 360 ppi it becoms difficult to tell them apart, but it can be done by close inspection." This confuses me. If there is a "distortion" or "softening" of some sort tied back to the aperture choice (or the use of post-processing correction in the f/16 example), it doesn't seem to me that printing at a higher resolution would improve that in any way, since those issues are optical and not related to print resolution. Especially it does not seem that whatever change this created would benefit an image that was supposedly more damaged by diffraction blur and/or distortion correction in post - how would increasing print resolution do that?
If the difference between the samples decreases as you increase print resolution, isn't that telling you something about your post-processing workflow and some potential difference between printing resolutions rather than apertures at the time of exposure? I'm assuming that the difference is that both of your 20" x 30" equivalent samples are otherwise processed the same way with the exception of using a print resolution of 360 instead of 240. If that is the difference, it seems that it could not be explained by anything in the camera or the capture or even the post-processing, right? (Unless your different sharpening methods for the two samples somehow play into this.)
Or maybe I'm misunderstanding... It has happened before.