Yes, both are on the same machine, and neither have proxy settings configured.
However - on FF I did have the 'auto-detect proxy settings' box ticked, rather than the 'Direct connection to the Internet' box. Changing it from the former to the latter has fixed the problem.
The return header no longer shows the hit/miss entry from the SME Server proxy:
HTTP/1.x 200 Document follows
Date: Wed, 27 Feb 2008 23:56:48 GMT
Server: MiniServ/0.01
Connection: close
Set-Cookie: testing=1; path=/; secure
Pragma: no-cache
Expires: Thu, 1 Jan 1970 00:00:00 GMT
Cache-Control: no-store, no-cache, must-revalidate, post-check=0, pre-check=0
Content-Type: text/html
I've checked my settings and the transparent proxy is turned off, which means that a browser can choose to use the proxy or not (FF was using it, and IE was not). That explains the difference.
However - that still leaves me with the problem that going through the proxy causes the page to fail. Obviously I'm fixed up now (I have my workaround) but I'm still curious as to what could be going wrong.