Here's a simple example of image upscaling using a noise free source image. This example shows the difference between sharpening an upscaled low resolution single image and applying super resolution to multiple low resolution images with subpixel shifts. For this example I wrote my own simple super resolution script in MATLAB. The image is first downscaled to half the resolution, than upscaled two times to the original image size (using bicubic).
The reality of sharpening is that it cannot retrieve details below the pixel level. This is where the difference with super resolution begins. Super resolution does retrieve details below the pixel level.
The first image shows the original image vs the downscaled + upscaled image:
http://screenshotcomparison.com/comparison/125890
Because the stripes on the girl's pants are at the pixel level, the low resolution image cannot resolve this detail, so it is lost. This lost detail cannot be retrieved by sharpening as is obvious from the next comparison:
http://screenshotcomparison.com/comparison/125891
The information about the stripes is lost in a single image, but can be retrieved by combining the information in multiple images. This is the strength of super resolution, as can be seen in the next comparison:
http://screenshotcomparison.com/comparison/125895
So, although there is overlap between sharpening and super resolution at the super pixel level, super resolution is the only way to retrieve details at the subpixel level. As such super resolution is generally more detailed, more accurate at reconstructing shapes, and has less artifacts, than sharpening. Therefore, if you have a super resolution algorithm available, it makes no sense to use sharpening, as you get all the advantages of sharpening with super resolution anyway.