Actually, you never enter your login details on anything other than twitter.com's own pages, and the token you get isn't an encoded username:password string.
Twitter has changed their authentication methods to a rather complex system of tokens and encryption, which is pretty much beyond the Arduino's capabilities. Instead, the current library connects to a webapp on the author's website which posts to Twitter on your behalf. It never knows your login details, it just asks Twitter.com to authenticate you and gets a special token from them that lets it post to your account, but not log in or change any settings. I expect the webapp doesn't save any of those tokens, but uses the one you pass in directly. Any other webapp would have to get their own individually authenticated tokens, as tokens are specific to each registered webapp.
Changing your password invalidates all such tokens, so you would have to get new ones after doing so.
(I have to wonder whether the webapp is able to handle the load of all the arduino's tweeting to it; I'm only getting very sporadic success with it right now.)