I could save my own copy and have my webapp always use that, but that's not such a great idea as when twitter change their API, they also change widget.js to compensate.
So my webapp always pulls in the master authoritative copy, but then I tend not to notice when it changes - and sometimes this breaks my assumptions about how it works (also, changes in widget.js are a good way to track changes to certain API details that I access outside of the widget.js library itself).
So, as there's also no RSS feed about updates or similar, I instead store widget.js in my source repository, even tho my app doesn't use the copy I've stored, and in my development workspace I regularly regularly fetch the latest widget.js (for me, this is in the script to restart my personal development environment, launch the webserver etc but you could do it in a cron job)
And thanks to the wonders of distributed source code control systems and their "no check out required" model (I use bzr, the same is true of git and others) then if the authoritative version hasn't changed, then this has no effect, but when it does change, I suddenly see the file as modified in my "modified files" report ("bzr status"), and I can diff the new against the old and see if I need to react accordingly.
It's more of a hack than anything else, but works surprisingly well, and has stopped me being caught out (especially with "silent" changes where there's no official communications about such things).