Well since I wrote that, I had to take my copy of the game back to the Target store that I bought it from. Why? 'Cuz my copy of the Windows version didn't have the white sticker inside the box that had the Windows LIVE Access Key printed on it for that particular DVD. So I uninstalled Fallout 3, exchanged the original copy for one that did have the sticker with the key numeral, and installed the game fresh.
So everything was cool... except that the game "hung" shortly after the birth/character creation scene started, and would go no further.
Some reading on the Intertubes showed that a lot of people have been having this problem also.
Well, this morning I worked at it a bit, and I found a solution. So I thought it'd be a good thing to share it with others who are also going through this...
If you are playing Fallout 3 on Windows Vista or XP, uninstall the game. Make sure that the Bethesda Softworks folder that's within your Program Files folder is deleted also.
Now here's the messy part that requires a bit of daring: use Regedit (you can run it from the Start button) and do a search in your Windows registry for all entries containing "Bethesda" or "Fallout3". This is what I figured had snagged me: my computer was having an "identity crisis" as to which copy of Fallout 3 it was supposed to be running. So scour your registry and carefully delete anything that refers to Bethesda Softworks or Fallout 3.
When that is done, close out Regedit. Then reboot Windows.
You should now have a fresh, pristine machine on which to re-install Fallout 3, that so far as it's concerned there'll be no evidence that the previous install ever happened. Since going through this procedure I haven't had any further problems with Fallout 3: it's running perfectly, and I also ran the update to version 126.96.36.199, again without any trouble.
Hope this helps some folks out. Now go gird up and get ready to take the Capital Wasteland by storm :-)