Well, I said for a long time that once I got a 5D III, I'd do some comparison shots. I've long held the opinion that crop sensor cameras, like the 7D, do have value in certain circumstances. The most significant use case where a camera like the 7D really shows it's edge over full frame cameras is in reach-limited situations. A reach limited situation is one in which you cannot get physically closer to your subject, and your subject does not fill the frame. The likely case is that you are using your longest lens, and will likely crop in post.
In the past, others have made the argument that a camera like the 5D III or 1D X has so much more image quality than a camera like the 7D that the 7D could never compare. The argument was made that an upsampled 5D III or 1D X image (or even, for that matter, D800/E, D600, etc. image) would be just as good.
I'd like to prove my case. I've taken the most reach-limited scenario possible...photographing the moon, with a 1200mm lens (Canon EF 600mm f/4 L II w/ Canon EF 2x TC III). I used a Canon EOS 7D and a Canon EOS 5D Mark III for imaging. The lens and camera were attached to an Orion Atlas EQ-G equatorial tracking mount, operating in Lunar tracking mode, to minimize any other factors that might affect image quality. Seeing (atmospheric turbulence measure) was average.
Above is a GIF image of the 7D and 5D III images scaled to the same size, overlaid directly on top of each other using Photoshop's layer difference blending mode for best possible alignment. Both images were created exactly the same way, by initially focusing with BackyardEOS' focus module for optimum focus (BYEOS is like having a 2560x1600 live view screen...it's awesome!) The image exposures for both cameras are 1/100s f/8 ISO 200. Five images for both cameras were taken, the best frame from each set was chosen for comparison. Both images were maximally cropped simply by choosing 1:1 in Lightroom. Both images had identical processing applied in Lightroom (one image was processed, it's settings were copied and pasted onto the other.) Both images were initially scaled to approximately 1/4 their original size (770x770 pixels, to be exact).
The 5D III image was then layered onto the 7D image, and upsampled in Photoshop by a scale factor of exactly 161.32359522807342533660887502944%. This scale factor was derived by computing the sensor diagonals of both cameras:
Then dividing the FF diagonal over the APS-C diagonal:
Then finally multiplying by 100% (to get a relative scale factor that I could directly apply with Photoshop's layer scaling tool.)
I believe the GIF above speaks for itself. The larger pixel size of the 5D III clearly does not resolve as much detail as the 7D does. Not only is the 7D image sharper, but there is a significant increase in fine details, small craters, nuances of color, etc. Here is another GIF, this time the images are only 1/2 original size (any larger, and the effects of seeing diminish any real benefit...I've had days where seeing is excellent, and more detail can be resolved, but sadly tonight was not one of those days):
The 7D's smaller pixels, despite being a generation prior to the 5D III's, are still resolving more detail, especially fine edges to crater rims (some of which don't even show up at all in the FF image), and are extracting a finer and more nuanced level of color. Many smaller craters, especially those that are inside larger craters, as well as the central mounds of many craters, are either difficult to make out or simply don't appear in the 5D III image, where as they show up clearly in the 7D image.
A common reach-limited use case is bird photography. Similar to the moon, it can be difficult to get close to and fully extract all the detail from a small songbird, shorebirds, and shy waders or waterfowl. One either needs a significantly longer lens on the full frame (I am still experimenting with the 5D III, but I'll probably be using 840mm and 1200mm a lot more than 600mm), or you need the skill to get much closer to your subjects, in order to fully take advantage of the benefits the larger frame has to offer.
Anyway, there you have it. The 5D III is an excellent camera, and when you have the option of framing identically (i.e. filling the frame with your subject), the larger frame trounces the 7D in terms of image quality. It gathers 2.6002949408613476991603214253469x more light:
With more than two and a half times more light, it's two and a half times better. Like using two and a half stops lower ISO on the cropped sensor. However if you don't have the option of either getting closer to your subject, or using a super long lens (not everyone has the option of spending $13,800 ($12,800+$500+$500) on a 600mm f/4 II and both of Canon's Mark III TCs), then there is no question that a camera like the 7D, or currently the better option the 70D, is going to give you the option of creating more detailed photos.
UPDATES:
Ok, here are a few updates, as per requested.
The first image here is the 7D and 5D III at "native" size. To further clarify my procedure from above. These images were "cropped", however they were cropped such that 100% of the sensor height was used. The only parts of the image that were discarded were the empty black sky areas to the left and right of the moon. That means, from a technical standpoint, these are 1:1 crops. They are then downsampled, but since I used 100% of the sensor height, these crops are directly indicative of the relative size difference of the moon in both frames.
You'll notice the 5D III image is sharp. Both images were sharp, or at least, as sharp as I could get them. I basically used a live view method of focusing, however one that is much more advanced. I used the program BackyardEOS, which is an astrophotography imaging tool that is specifically designed for Canon EOS cameras (which are endemic in the astrophotography world for budget imagers...the T3, T3i, 60Da and 6D are pretty much the top cameras you'll find in astrophotographers kits...those that don't use dedicated astrophotography CCDs.)
BYEOS has a brilliant frame and focus wizard. It takes the live view feed from the camera, and renders it on a computer screen. I can maximize the program and basically get 2560x1600 live view (minus a bit program panels and border).
I use these tools to focus:
I use coarse and medium to get focus close, then step with fine. The fine focusing arrows are extremely fine...they are designed to focus stars, so they move the focus group in the lens by the smallest possible amount. I spent about five minutes with these tools with both cameras to find the best focus possible. It isn't as easy as it sounds...you don't just end up with a crisp, sharp moon. The moon, at that size, looks more like it was dropped into a vat of boiling water, and every few seconds you have a moment where the "water" (atmospheric turbulence or "seeing") clears, and in that moment you have to gauge whether to focus forward or back to get it better.
So, the images are focused as best as I could get them.
The next image here is a noise comparison. It has three frames...a 7D crop that is unscaled, a 5D III crop that is unscaled, and a 7D crop that mirrors the 5D III crop that IS scaled. The 7D, at native size, definitely has more noise. It also looks almost as soft as the 5D III image.
When the 7D image is downsampled to the same size as the 5D III image...any advantage the 5D III had in terms of noise disappears. The 7D image clears up a bit, and appears a little sharper. Fine details pop a little bit more than they do with the 5D III.
Why? Because the moon covered the same absolute sensor area. There is a difference in pixel count between the two images, but overall, both sensors gathered exactly the same amount of light! That's the key there. There is no advantage to a larger sensor if you are not utilizing that increase in sensor area. If your using the same exact absolute sensor area between both cameras...there is no difference. If the 7D had 6.25µm pixels, then the two cameras, in this kind of situation, would perform IDENTICALLY.
In a reach-limited scenario, you want smaller pixels. It really doesn't matter if your using a full frame sensor, a medium format sensor, a micro 4/3rds sensor, or an APS-C sensor. If the pixels are all the same size, and you put the same number of pixels on your subject...assuming all four of those sensors use the same technology, there is literally no difference. That is usually not the case, though. Smaller sensors generally tend to use smaller pixels. The 7D still has smaller pixels than the D800 and D810. Smaller pixels trump bigger pixels when you are reach-limited.
In the past, others have made the argument that a camera like the 5D III or 1D X has so much more image quality than a camera like the 7D that the 7D could never compare. The argument was made that an upsampled 5D III or 1D X image (or even, for that matter, D800/E, D600, etc. image) would be just as good.
I'd like to prove my case. I've taken the most reach-limited scenario possible...photographing the moon, with a 1200mm lens (Canon EF 600mm f/4 L II w/ Canon EF 2x TC III). I used a Canon EOS 7D and a Canon EOS 5D Mark III for imaging. The lens and camera were attached to an Orion Atlas EQ-G equatorial tracking mount, operating in Lunar tracking mode, to minimize any other factors that might affect image quality. Seeing (atmospheric turbulence measure) was average.

Above is a GIF image of the 7D and 5D III images scaled to the same size, overlaid directly on top of each other using Photoshop's layer difference blending mode for best possible alignment. Both images were created exactly the same way, by initially focusing with BackyardEOS' focus module for optimum focus (BYEOS is like having a 2560x1600 live view screen...it's awesome!) The image exposures for both cameras are 1/100s f/8 ISO 200. Five images for both cameras were taken, the best frame from each set was chosen for comparison. Both images were maximally cropped simply by choosing 1:1 in Lightroom. Both images had identical processing applied in Lightroom (one image was processed, it's settings were copied and pasted onto the other.) Both images were initially scaled to approximately 1/4 their original size (770x770 pixels, to be exact).
The 5D III image was then layered onto the 7D image, and upsampled in Photoshop by a scale factor of exactly 161.32359522807342533660887502944%. This scale factor was derived by computing the sensor diagonals of both cameras:
Code:
ffDiag = SQRT(36^2 + 24^2) = 43.266615305567871517430655209646
apscDiag = SQRT(22.3^2 + 14.9^2) = 26.819768828235637870277777227866
Then dividing the FF diagonal over the APS-C diagonal:
Code:
43.266615305567871517430655209646/26.819768828235637870277777227866 = 1.6132359522807342533660887502944x
Then finally multiplying by 100% (to get a relative scale factor that I could directly apply with Photoshop's layer scaling tool.)
I believe the GIF above speaks for itself. The larger pixel size of the 5D III clearly does not resolve as much detail as the 7D does. Not only is the 7D image sharper, but there is a significant increase in fine details, small craters, nuances of color, etc. Here is another GIF, this time the images are only 1/2 original size (any larger, and the effects of seeing diminish any real benefit...I've had days where seeing is excellent, and more detail can be resolved, but sadly tonight was not one of those days):

The 7D's smaller pixels, despite being a generation prior to the 5D III's, are still resolving more detail, especially fine edges to crater rims (some of which don't even show up at all in the FF image), and are extracting a finer and more nuanced level of color. Many smaller craters, especially those that are inside larger craters, as well as the central mounds of many craters, are either difficult to make out or simply don't appear in the 5D III image, where as they show up clearly in the 7D image.
A common reach-limited use case is bird photography. Similar to the moon, it can be difficult to get close to and fully extract all the detail from a small songbird, shorebirds, and shy waders or waterfowl. One either needs a significantly longer lens on the full frame (I am still experimenting with the 5D III, but I'll probably be using 840mm and 1200mm a lot more than 600mm), or you need the skill to get much closer to your subjects, in order to fully take advantage of the benefits the larger frame has to offer.
Anyway, there you have it. The 5D III is an excellent camera, and when you have the option of framing identically (i.e. filling the frame with your subject), the larger frame trounces the 7D in terms of image quality. It gathers 2.6002949408613476991603214253469x more light:
Code:
(36 * 24) / (22.3 * 14.9) = 864 / 332.27 = 2.6002949408613476991603214253469
With more than two and a half times more light, it's two and a half times better. Like using two and a half stops lower ISO on the cropped sensor. However if you don't have the option of either getting closer to your subject, or using a super long lens (not everyone has the option of spending $13,800 ($12,800+$500+$500) on a 600mm f/4 II and both of Canon's Mark III TCs), then there is no question that a camera like the 7D, or currently the better option the 70D, is going to give you the option of creating more detailed photos.
UPDATES:
Ok, here are a few updates, as per requested.
The first image here is the 7D and 5D III at "native" size. To further clarify my procedure from above. These images were "cropped", however they were cropped such that 100% of the sensor height was used. The only parts of the image that were discarded were the empty black sky areas to the left and right of the moon. That means, from a technical standpoint, these are 1:1 crops. They are then downsampled, but since I used 100% of the sensor height, these crops are directly indicative of the relative size difference of the moon in both frames.

You'll notice the 5D III image is sharp. Both images were sharp, or at least, as sharp as I could get them. I basically used a live view method of focusing, however one that is much more advanced. I used the program BackyardEOS, which is an astrophotography imaging tool that is specifically designed for Canon EOS cameras (which are endemic in the astrophotography world for budget imagers...the T3, T3i, 60Da and 6D are pretty much the top cameras you'll find in astrophotographers kits...those that don't use dedicated astrophotography CCDs.)

BYEOS has a brilliant frame and focus wizard. It takes the live view feed from the camera, and renders it on a computer screen. I can maximize the program and basically get 2560x1600 live view (minus a bit program panels and border).
I use these tools to focus:

I use coarse and medium to get focus close, then step with fine. The fine focusing arrows are extremely fine...they are designed to focus stars, so they move the focus group in the lens by the smallest possible amount. I spent about five minutes with these tools with both cameras to find the best focus possible. It isn't as easy as it sounds...you don't just end up with a crisp, sharp moon. The moon, at that size, looks more like it was dropped into a vat of boiling water, and every few seconds you have a moment where the "water" (atmospheric turbulence or "seeing") clears, and in that moment you have to gauge whether to focus forward or back to get it better.
So, the images are focused as best as I could get them.
The next image here is a noise comparison. It has three frames...a 7D crop that is unscaled, a 5D III crop that is unscaled, and a 7D crop that mirrors the 5D III crop that IS scaled. The 7D, at native size, definitely has more noise. It also looks almost as soft as the 5D III image.

When the 7D image is downsampled to the same size as the 5D III image...any advantage the 5D III had in terms of noise disappears. The 7D image clears up a bit, and appears a little sharper. Fine details pop a little bit more than they do with the 5D III.
Why? Because the moon covered the same absolute sensor area. There is a difference in pixel count between the two images, but overall, both sensors gathered exactly the same amount of light! That's the key there. There is no advantage to a larger sensor if you are not utilizing that increase in sensor area. If your using the same exact absolute sensor area between both cameras...there is no difference. If the 7D had 6.25µm pixels, then the two cameras, in this kind of situation, would perform IDENTICALLY.
In a reach-limited scenario, you want smaller pixels. It really doesn't matter if your using a full frame sensor, a medium format sensor, a micro 4/3rds sensor, or an APS-C sensor. If the pixels are all the same size, and you put the same number of pixels on your subject...assuming all four of those sensors use the same technology, there is literally no difference. That is usually not the case, though. Smaller sensors generally tend to use smaller pixels. The 7D still has smaller pixels than the D800 and D810. Smaller pixels trump bigger pixels when you are reach-limited.