Having a bit of a task using Apple Mail with Google Apps Gmail. Every few days it insists on me signing in to the web interface, before Apple Mail can get access to my Inbox. Why is this?
Check your Keychain (Applications -> Utilities -> Keychain Access).
You can run Repair on the Keychain, which may be overkill, or just delete the info for your Google Apps and then sign in to see if that helps. I would suggest closing Mail before doing either of the two.