Killing ends when the thing you are killing (your sworn enemy, all hope, a pesky mosquito) is dead. Annihilate goes farther—when you annihilate something, you wipe all trace of it from the earth.
You kill a person, but you annihilate a tribe, a town, or even a species. It's nice to think that the human race made it all the way up until the 16th century before annihilate came to mean what it does now. Might this have something to do with the increasing use of gunpowder around this time?