At long last I've found out what British Telecom claims
I spoke to one of their Softphone 'advisers' who said, "It'll be to load the drivers for the Agile 'phone."
Although I'm not too keen on the "It'll be" part, it seems BT has a telephone called Agile that plugs directly into a USB port and the BTAgile.exe Startup entry is therefore only helpful if you have an Agile 'phone AND you ALWAYS want to have it activated when you bootup. So;
** If you DON'T have an Agile 'phone - you can safely delete BTAgile.exe from your Startup.
** If you DO have an Agile 'phone, but don't want it activated until you choose to do so - delete BTAgile.exe from your Startup, but find it in the Program Files, right click, create a Shortcut, and place the Shortcut in a convenient place such as the Desktop or (like me) a desktop folder called BT Utilities (where I keep all their stuff) so you can activate it at your pleasure.
** If you DO have an Agile 'phone and want it activated all the time from bootup - leave the Startup entry alone.
The above info needs to be added (in some form or another) to this page:http://www.bleepingcomputer.com/startups/BTAgile-19547.html
Such a pity BT is so reluctant to part with this information (it took me over 6 months) but I hope the above helps.