Logical and evidential arguments from evilLogical arguments from evil seek to show that God's existence is logically incompatible with the existence of (certain) evil. That is, it is argued to be logically impossible that God exist alongside such evil. The arguments then derive God's nonexistence from the existence of such evil. The classic proponent is J. L. Mackie.
Evidential arguments from evil don't allege that God and such evil cannot coexist. Instead, they merely seek to show that belief in God's nonexistence is reasonable, given the existence of such evil. They usually rely on claims about our epistemic situation regarding such evil (e.g., we can't see how it could be morally justified). The classic proponent is William Rowe.
It is important to distinguish between metaphysical possibility and merely epistemic 'for all we know' possibility. When philosophers say, "It's possible that God exists alongside evil," they commit themselves to the claim that the world might have had both. They might say, "There is a possible world with both". They don't mean merely, "For all I know, God really does exist alongside evil".
It is a (sometimes challenged) consensus among philosophers of religion that the logical arguments from evil are dead -- most people think they are not worth defending. But, is important to realize, this is not because everyone now agrees that God's existence is compatible with evil. On the contrary, many would agree that such compatibility is unproven. Rather, the obituaries stem from the new consensus that the incompatibility claim is also unproven. The problem is that it is difficult to judge the incompatibility claim one way or the other.
Alvin Plantinga, it is true, gave a much heralded argument for compatibility. This was his "free will defense". But the argument relies on a good deal of suspect metaphysics. It is quite reasonable to doubt that the mechanisms underlying his argument are really (metaphysically) possible. Even sympathetic philosophers now conclude that the compatibility claim is unproven; Plantinga's scenario might, for all we know, sketch a possible world, but it might not. We don't know one way or the other.
As parenthetically noted above, some philosophers (e.g., Richard Gale, Quentin Smith, Jordan Sobel) challenge the alleged death of logical arguments from evil. They attempt to support the incompatibility claim anew, sometimes relying on inductive considerations more commonly associated with evidential arguments from evil. Nevertheless, most philosophers discussing evil as support for atheism are more interested in evidential arguments from evil.