This is a good exercise, and there are some interesting differences between these models which are illustrative of things worth comparing.

• Technology generations: The Fujifilm camera is from 2010 and the Canon model from two years earlier. Electronics continue their march of getting cheaper and faster, and in general newer models have an advantage — although this is most true in the midrange. At the high end, more expensive initial choices give cameras longer functional lives (and non-electronic benefits like better controls and better build aren't influenced by tech improvements); at the low-end, new models might "spend" the improved tech on lowering costs rather than increasing quality.
• Image stabilization: The Canon camera in your example has a "real" optical stabilization system, while the Fujifilm camera says it has "Digital Image Stabilization" which is doublespeak-like industry code for "no image stabilization".
• Sensor size: The Canon camera has a 1/2.5-inch type sensor, while the Fujifilm has a slightly bigger 1/2.3-inch sensor — this is about 15% more surface area, which in this case is not very significant. But in some cases, the sensor size can really make a difference.
• Optics: Both models feature a relatively useful moderate zoom. There's no way to compare image quality from the specs, and that's something that's probably also worth looking into if you can. The Fujifilm has a more conservative 3× zoom range (which is usually better for image quality), and provides a more-useful slightly wider angle (but not by much). Since you have the cameras, some actual test shots would be useful.
• Aperture: Related; the Canon camera opens up to f/2.6 at the wide end and f/5.5 at the long end of the zoom range. The Fujifilm is a third of a stop slower at the wide end, and similar at the telephoto end. (It doesn't zoom out as far so direct comparison isn't easy.) This is a more significant difference than the 15% sensor area, but it's also in this case pretty small. In some cameras, the difference is worth making a big deal about.
• CHDK: There's nifty firmware hacks available for Canon P&S cameras, including this model; there's really nothing like it for other brands. That might be a way to make an older camera do some really cool new tricks.
• Viewfinder: The Canon has one, the Fujifilm does not. This isn't a through-the-lens finder (it's basically a little tunnel of plastic optics) and so is less useful than the one on a DSLR, but it may still fit your photography better.
• Control modes: Honestly, this is less useful on a small-sensor camera, but if you're learning about photography it's nice to be able to take more control. Neither camera offers "full manual", but the Canon offers shutter priority and aperture priority program modes, while the Fujifilm is auto-exposure only. (In fact, I'm not sure it even has EV compensation or exposure lock, which would be a big concern.)
• Responsiveness: You can't really tell this from specs, but maybe can get it from reviews. Testing it yourself is of course great. There's several important things to look for: time it takes to go from off to ready, time it takes from shot to ready again, and lag between when you fully-press the shutter to actually taking the photo. Secondarily, a fast AF speed and at least moderate burst rate are nice.

Then there's control layout, build, and general feel. This varies model to model but the different brands do tend to fit some generalization. For example, in every Canon P&S camera I've used — including the high-end ones — zooming is very clearly a "stepped" operation. You press the zoom lever, and it goes from 35mm to 50mm to 85mm, or whatever sequence of pre-programmed points the camera offers. Fujifilm cameras still use a servomotor that works in the same way (with the exception of a few which have the great feature of using manual zoom lenses), but there are more steps and it feels smoother.

Whether this is a concern or bother to you is somewhat personal — as is the issue of control or feel in general, but for learning photography, you do want as many accessible controls as possible, and basically more buttons is better, because fiddling in menus is no good.