I finally found the time to sit down and nut this thing out.
Richard, you were right, as you usually are! There were two common programs that were causing the problem but I was able to resolve it without in-place OS re-installs, etc.
I run Zone Alarm on both computers and had slightly different advanced configurations on them. I also run Webwasher, my ad blocker of choice, on both.
After a bit of research
here and tweaking Zone Alarm on both machines, I had WMP working on one but not the other. It was then I noticed that Webwasher was not running on the one where WMP was working. When I unloaded Webwasher (as distinct from just switching it off) off the other machine - presto, WMP worked on that one too. The IE misbehaviour with on-line multimedia content was also fixed.
I'm sure it's just a configuration issue between Webwasher/IE/WMP, and I'll chase that up at a later date.
In the meantime I'm just happy to have the problem resolved without having to rebuild OSes. Thanks for the tips, folks.
BTW, I found the link while browsing MS's WMP discussion group. I think Zach, whose website the link points to, is an MS staffer specialising in WMP.
AA
Edit: 5 hours, a bite of dinner and a glass or two of good Aussie red later, have fixed the Webwasher configuration. Yeehaa!