I've used both and don't see any difference in security but the newer 2 Factor Authentication does seem a bit buggy.
With the older 2 Step Verification you choose (from the device where you are wanting to gain access) each time where to send the code. 2FA however automatically sends a prompt asking whether to allow access to all trusted devices -- except that it sometimes doesn't send to all of them and I need to have it resend to get it to go to the trusted device that I have available.
Also, when I say that it sends to all devices, that literally means all, including the device that is prompting for the code -- and that is confusing at first. If you try to "allow" from the same device that is trying to gain access (and that would be a huge security issue if it worked as it would mean you don't need a second device) it does display a code on that device but entering the code won't actually give access.
All in all, it does work but it also seems to be a bit of a work in progress.