Here's an example:
In this image, there are two black disks on a white background. One of the disks is half a pixel higher than the other, and this is clearly visible. If I wanted to align the two, I could upsample both images to get higher-resolution but blurry (or in some other way degraded) versions of both. Aligning these would be easier, and then I could report what I did ("Move the second image half an original pixel down"). All of the various image registration programs (3dvolreg, FSL FLIRT, SPM realign, etc.) are designed to do something like this, although each of them has its own clever way to avoid actually upsampling the images, because doing so would take too much time and memory.