Difference between AT&T iPhone 5s/c and VERIZON/SPRINT iPhone 5s/c?

I'm looking to buy an iPhone 5s or c (haven't made up my mind yet) on eBay and I see that Verizon and Sprint iPhones are significantly easier to find and cheaper than AT&T iPhones. Why is that? What are the major differences that make these products of Verizon and Sprint cheaper?

Also, could anyone please tell me the difference between iPhone 5s and 5c please.

The different carrier versions of all iPhones are exactly the same, except for radio frequencies which don't connect to other carrier's towers. An iPhone 5S or 5C is the same on AT&T, Verizon, Sprint, T-Mobile, and literally any carrier in the world, save for the radio bands. Be careful, though. An AT&T phone won't work with Verizon or Sprint, but will work with T-Mobile and internationally IF CARRIER UNLOCKED. A Sprint or Verizon phone will NOT work on any other carrier than the one it it is branded under. They're probably cheaper with some carriers because iPhones are more popular with those carriers.

The difference between 5S & 5C is that the 5C is the exact same inside as an iPhone 5, but has a tacky plastic shell. The 5S has the fingerprint scanner, a better camera, a better build, dual flash, and will likely be updated for longer than it's cheaper brother. If color means A LOT to you, buy a 5S. Otherwise, if you can afford to spend an extra $100, definitely go with the 5S.