2-factor authorization is the newer system - Two-factor authentication for Apple ID - Apple Support It ties all your Apple trusted devices to your AppleID and iCloud account. It does require iOS 9 or newer and OS X 10.11 or newer to use.
2-step verification is older - Two-step verification for Apple ID - Apple Support and requires an SMS number to use. While you can register phone numbers with 2-factor authentication you do not need to as it uses Apple's iCloud service and iCloud push notifications and the find my iPhone location tracking system.
That in a nutshell is the main difference between the two - the newer system is wholly integrated with your iCloud account and those devices you designate as trusted. As such, it does not rely on any cellular telephone system for codes as it is entirely internet based (but can still send a code to a telephone number if necessary or as a backup source to receive codes). The older system was based around cellular SMS messages and did use the initial (pre-iCloud) find my iPhone system but was not tightly integrated into iCloud and it's tracking/notification and internet encrypted communication like the newer system now is.
So if your devices and OS support it, 2-factor authentication is what you should use. If you do only have one Apple device though, you'd be advised to register an alternate trusted telephone number as an emergency way to get verification codes. I use one of my google voice numbers for that, since I could get messages sent to that in any web browser (and my google accounts themselves are all 2-step login protected as well).