Now, one way to look at a PCB is to generate gerbers (the standard "plotter format" for PCBs), and use one of the gerber viewers to look at those. This has the advantage of simplifying the "picture", AND most gerber viewers will load up multiple layers of a PCB so that you can see them all together. I realized that it would also let you display multiple copies of the same layer, so that you could "flip" through them and (hopefully) notice changes. In fact, the gerber viewer I use (gerbv, an open-source utility that runs on Windows, Macs, and Linuxes), has a display mode where the layer are displayed with their colors XORed together. (In theory, this is an option to allow you to look at multiple layers at once, although in reality how well it works depends on the actual bit patterns of the colors used for each layer.)
However, if you DO pick your layer colors carefully, you can display two (or more) layers that are almost the same, and have the differences HIGHLIGHTED for you! This seemed close to ideal for visualizing the changes in a PCB design. Well, for small changes, anyway. We'll have to see how the community feels about its usefulness.
All it needed was some sort of script that would go through all the backup files, generate appropriate gerber files, and then load up the gerber viewer having given each file an appropriate set of color, color mixing mode, and other display parameters. It seems to be somewhat useful. Here are some examples of how it works.