It is important to distinguish between the problem of evil and the argument from evil. The first is the problem of reconciling the existence of God, as traditionally understood, with the existence of natural and moral evils. As J. L. Mackie points out, this "is essentially a logical problem: it sets the theist the task of clarifying and if possible reconciling the several beliefs which he holds." (The Miracle of Theism, Oxford 1982, p. 150) Mackie goes on to point out that "the problem in this sense signally does not arise for those whose views of the world are markedly different from traditional theism." Thus the theist's problem of evil does not arise for an atheist. It might, however, be the case that some other problem of evil arises for the atheist, say, the problem of reconciling the existence of evil with life's being worth living. But that is a separate matter. I discuss it in A Problem of Evil for Atheists.
The argument from evil, on the other hand, is an attempt to show the nonexistence of God from the fact of evil, where 'fact of evil' is elliptical for 'the existence of natural and moral evils.'
THE DIFFERENCE BETWEEN THE PROBLEM OF EVIL AND THE ARGUMENT FROM EVIL
The main difference between the problem of evil and the argument from evil is that the former is an ad hominem argument whereas the second is not. I am using ad hominem in the way Peter Geach uses it on pp. 26-27 of his Reason and Argument (Basil Blackwell 1976):
This Latin term indicates that these are arguments addressed to a particular man -- in fact, the other fellow you are disputing with. You start from something he believes as a premise, and infer a conclusion he won't admit to be true. If you have not been cheating in your reasoning, you will have shown that your opponent's present body of beliefs is inconsistent and it's up to him to modify it somewhere.
As Geach points out, there is nothing fallacious about such an argumentative procedure. If A succeeds in showing B that his doxastic system harbors a contradiction, then not everything that B believes can be true. Now can an atheist prove the nonexistence of God in this way? No he cannot: at the very most he can prove (with the aid of various auxiliary premises that he and his interlocutor both accept) that God exists and Evil exists cannot both be true. But it does not follow therefrom that God exists is not true. For the atheist to transform the ad hominem problem of evil into a non-ad hominem argument from evil, he would have to establish, or at least assert, that evil exists, and not merely that the theist believes that evil exists. To see my point consider the following conditional, where P is the conjunction of auxiliary premises:
C. If evil exists & P, then God does not exist.
The atheist who raises the problem of evil for the theist asserts (C), or rather a proposition of that form. But to assert a conditional is not to assert its antecedent, or its consequent for that matter; it is to assert an entailment connection between the two. Now although it is the case that for each argument there is a corresponding conditional, and vice versa, arguments must not be confused with conditionals.
Transforming (C) into an argument from evil yields:
God does not exist.
Clearly, an atheist who gives this argument, or rather an argument of this form, must assert both premises. Doing so, he ceases his ad hominem examination of the consistency of another person's beliefs, beliefs he either rejects or takes no stand on, and 'comes clean' with his own beliefs.
THE ARGUER FROM EVIL NEEDS TO AFFIRM OBJECTIVE EVIL
If the atheist's aim is merely to poke holes in the logical consistency of the theist's belief set, then it doesn't matter whether he thinks of evil as objective or subjective. Indeed, he needn't believe in evil in any sense. He could hold that it is an illusion. But if the atheist's goal is to support his own belief that God does not exist with an argument from evil, then he needs to maintain that evil is objective or objectively real.
Consider all the enslavement of humans by humans that has taken place in the history of the world. Suppose it is agreed that slavery is morally wrong. What makes this true? Define a moral subjectivist as one who agrees that the claim in question is true, but holds that the truth-maker of this moral truth, and of others like it, is an individual's being in a psychological state, say, the state of being repulsed by slavery. For the moral subjectivist, then, sentences like 'Slavery is wrong' are elliptical for sentences like 'Slavery is wrong-for-X,' where X is a person or any being capable of being in psychological states. Furthermore, the moral subjectivist grants that moral claims have truth-makers, indeed objective truth-makers; it is just that these truth-makers involve psychological states that vary from person to person.
Now if our atheist subscribes to a theory of evil along those lines, then, although there will be objective facts of the matter regarding what various individuals feel about the practice or the institution of slavery, there will be no objective fact of the matter regarding the wrongness or moral evil of slavery.
If so, the fact of evil subjectively construed will have no bearing on the existence of God, a fact, if it a fact, that is objective.
Suppose a torturer tortures his victim to death solely for the satisfaction it gives him. And suppose that moral subjectivism is true. Then the torturing, though evil for the tortured, is good for the torturer, with the upshot that the torturing is neither good nor evil objectively. Now if I were on the scene and had the power to stop the torturing, but did not, would my noninterference detract from my moral goodness? Not at all. (The same goes a fortiori for God.) For nothing objectively evil is transpiring: all that is going on is that one person is securing his pleasure at the expense of another's pain. If you insist that something evil is going on, then that shows that you reject moral subjectivism. But if you accept moral subjectivism, then nothing evil is going on; the torturing is evil only in the mind of the victim and in the minds of any others who sympathize with him. If you accept moral subjectivism and continue to insist that the torturing is evil, then you would also have to insist that it is good, since it is good from the perspective of the torturer. But if it is both good and evil, then it is (objectively) neither.
What I am claiming, then, is that the atheist arguer from evil must construe evil objectively. This will result in trouble for the atheist if it can be shown that objective evil cannot exist unless God exists. For then the atheist arguer from evil will end up presupposing the very being whose existence he is out to deny. No doubt this is a big 'if.' But it is worth exploring. The problem for the atheist is to explain how there can be objective good and evil in a Godless universe. I wish him the best of luck with that.
And another line worth exploring is a theistic argument to God from the fact of objective good and evil. No such argument could PROVE the existence of God, but it could very well have the power of cancelling out the argument from evil.