In my opinion, the DDR 333Mhz module should work on the old
mainboard, it is just that it will run at a lower speed that is
limited by the motherboard. So if your mainboard can support RAM
speed up to 266Mhz, than the DDR 333Mhz module will only run at
266Mhz.
You can download a bootable program to test your memory.
http://www.memtest86.com/
Recommended: download the ISO file to create a bootable CD.
Then boot from CD to run some memory test.
If you want to be more professional, you can try this:
http://www.uxd.com/rstpro2.shtml
This is definitely better since you can even pinpoint which part of
the memory has failed. (However, unless you are into a computer
business, it is unlikely you will purchase this testing
device.)