Also confirming that I have encountered this problem.
I use Google Apps for Work to host my domain for personal use and I'm the only super admin user. Across Google's support pages it states:
"If you have iOS 8.3 on your iPhone or OSX 10.10.3 on your Mac, you will no longer have to use App passwords to use 2-Step Verification."
However, this doesn't seem to be the case for Google Apps users - I continually get a "Failed to authenticate with Google" pop-up every time I attempt to add my Google Internet Account under "Internet Accounts" on Mac OS 10.3.3, whereas the whole procedure simply quits under iOS 8.3.
It worked before and I've made no changes within my Google Apps admin console, so something has changed, somewhere!
I have also tested this with Google's 2-step verification disabled and I get the same issues. Interestingly, I have no problem adding a standard Google account to the same Mac and iPhone, with and without 2-step verification.
These steps lead to the issue:
iMac OSX 10.3.3:
1. I go to Systems Preferences and select Internet Accounts.
2. I click on the Google logo
3. I sign in with my Google Account
4. I use Google Authenticator to verify (I have also tried this with verification turned off, but still get the same error)
5. I then get a window stating that OS X would like to view and manage my mail, email address, basic profile info, calendars, messages and contacts.
6. I click Accept
7. I receive a pop-up stating "Authentication failed, failed to authenticate with Google."
iPhone iOS 8.3:
1. I go to Settings, then "Mail, Contacts, Calendars" and select "Add account"
2. I again click on the Google logo and repeat steps 3 to 6 as above.
3. My iPhone then quits the process with no error message at all and no account has been setup.
As I stated earlier, I do not have this problem with a standard Google Account, so it seems to be an issue related to Google Apps for Work users.
As others have mentioned, I can only setup my iMac's Mail App's access to Gmail's IMAP servers by selecting "Add Other Account..." either in Internet Accounts or Mail's Preferences > Add Mail account, and then using the App-Specific Password method. However, doing this with my iPhone, means I have to create separate accounts and separate App-Specific Passwords for both Gmail and Google's CalDAV Calendar, rather than have a unified "single sign-on" Google account. So not only am I still relying on App-Specific Passwords when Google states "I will no longer have to use App passwords to use 2-Step Verification.", it also makes for a messy setup and means my iPhone's mail folders behave differently, especially when it comes to Archiving/All Mail options.
I'm currently wondering if it's an issue with Google Apps for Work accounts and Apple’s implementation ofGoogle Sign-In and/or Google’s OpenID, the latter of which was deprecated by Google on April 20th, which is why I am here hoping to find a solution.
I've also raised the issue with Google's Support Team, but they seem to be sending me down the Google/Mobile Sync route using a MS Exchange server, which is not how I want things setup.
Anyway, I hope Apple and Google can work together on this one, as I feel there's a few of us here with this problem...