YN 560 is a manual flash. That means either through judgement, trial/error, light metering measurements, u adjust the flash power to give u the appropriate lighting for your photo.
580 EX II is a ETTL flash. Canon's top flash model which will calculate the appropriate lighting for you, when mounted on most new Canon DSLRs.
Only owned the YN560 so i'll share some of my thoughts.
YN 560 being sold by alot of sellers in the 'Mass Sales' section. Prices around the 120 range i think.
It is powerful. So powerful that I hardly even use the maximum flash power most of the time o.o
It feels sturdy and looks well designed.
It is extremely prone to reliability issues. I was skeptical about this but I experienced it first hand when my 1st generation model(plastic mount) zoom motor failed. Used sparingly fired only about 200 shots without even moving much of the zoom head. I got quite edgey cos i only bought it brand new for 2 weeks. Being itchy-handed, I opened it up and managed to fix it.
The new batch apparent had better quality control. Meanwhile, with a working YN 560, I'll say its quite bang for the buck. However, do note that I think its quite risky, especially for newbies, to pay 120-ish dollars on a product that is notoriously prone to failure. Tons of criticism online. U can google them to find out more.