Is it possible (and safe/reliable/non-damaging) to use a 20-pin ATX PSU with a 24-pin ATX motherboard?
Solution 1:
I have no exact information to provide you here, but my gut feel would be that you should only consider trying this if you have a relatively low-power system on the motherboard. Something with not too much memory, ideally no more than 1 hard-disk, and probably not too high-end a dual-core processor... if you have any kind of modern graphics card installed I would probably not even consider going there.
Note that what the 24-pin supply does, is add one additional supply line of +3.3V, +5V and +12V each. If you use the adapter you suggest it will copy one or more of the existing lines to supply these pins... which means that they need to share the additional load.
From having a look at the Wikipedia page you reference, it looks like the standard 20 pins have a decent number of +5V and +3.3V lines already, but there is only one +12V line, which gets duplicated by the extra 4 pins. I do not know offhand what hardware uses the +12V, but that would be my main stability concern when using the splitter attachment.
Solution 2:
I used a 20 pin power supply on a motherboard with a 24 pin socket for years with no problems at all.
I had forgotten all about it until this last weekend when I replaced the PSU (failing fan) and had to remove a sticker that the manufacturer had placed over the last 4 pins (with a little graphic indicating that they were needed only with a 24 pin PSU).
This was with an AMD Athlon64 X2 4800+ CPU, mobo video, 4GB RAM in 2 sticks, 3 hard disks, DVD writer and commonly running 7 USB devices, 6 of which are powered from the computer.
In retrospect, I am a little surprised.