Double ropes are independently rated to take a leader fall. That is, one strand of the rope can support a fall(s). You use them by clipping protection alternately, or at least in a way that reduces rope drag.
Twin ropes are often a bit thinner, and are NOT rated to take a fall on a single strand. They should both always be clipped to every piece of protection, just like a single rope.
In the central/eastern alps, twin ropes are more popular than double ropes. My double ropes are about ready to retire and I think I'll replace them with twins. They are really convenient, as long as you can avoid twisting them in knots at belays.
No matter what I tell my various partners about alternating clipping, they just always clip both ropes to each piton. May as well just make it official.
Double ropes are great on trad terrain, especially where the lines wander. You can save a LOT of slings and rope drag, as the two lines can end up several meters apart from each other on a wandering pitch. No problem.