If you don't fully pay it off, yes, you have to turn in the phone. If they're like AT&T, you can pay off the remaining balance, but it's going to be a good chunk after only 12 months I assume. For an S6, you'd have to pay $300 after the 12 month mark to keep the phone.
If you want a new phone every year, you might be better off with traditional contract plans. In the off year, you're gonna have to just sell your current phone and pay full price for a new one. That would save you $100 over 12 months ($300 for Edge plan at the end of 12 months to keep it, vs. $200 now to buy it on contract).
This is all assuming you're on a data plan that charges $15 for Edge smartphone lines. If you're on a lower plan (4GB or less), you only get a $15 discount for Edge, as opposed to $25. In that case you're definitely better off with contract option.
edit: Edge would be "cheaper" if you aren't the type to bother selling your phones, but then you'd have to get into the mindset that you're renting a phone for 12 months. You'd never have to put any money down for a phone (except taxes up front). That's not really a mindset I can get into as I like keeping/selling my old phones, but that's the option they give.
edit2: Damn, I just noticed Verizon is cheaper than AT&T. Crazyness!