Same problem here after 10.10.3 came along. Getting errors from "Internet Accounts" with 2-step Google verification, but only on one of my three Google accounts. I have a regular Google account (working fine) and two Google accounts set up via "Google Apps" (both from employers; one works and one doesn't).
I enter my password into "Internet Accounts", which it seems to accept since it then sends me a 2-step verification code to my cell phone. Once I enter that code, it accepts it and brings me to a screen that says "OS X would like to: View and manage mail, profile info, etc.". Clicking "Accept" on that yields: "Authentication failed - Failed to authenticate with Google."
Mac Mail still works with all three accounts, fortunately, but the error keeps coming up in "Internet Accounts."
The account that is having problems is the only one of the three where I use a different e-mail address than my login with Google. In other words, my login is (for example): "fred@abc.edu" but the e-mail address I use is "fred@mail.abc.edu". Of course, I have no idea if that has anything to do with it.