Two-step verification vs two-factor authentication

What's the difference between two-step verification and two-factor authentication?

I've tried to find the answer online, but all I've discovered so far is that two-factor authentication is the newer and better version of two-step verification.

As far as I can tell, they both work the same way (log on, get a code on your iDevice, enter that code on the device trying to log on.

Can someone please clarify?

Thank you.