When we look at the fact of evil in the world, which worldview best accounts for that fact—Theism or Atheism?

I admit that both options are difficult, but atheism fares worse than theism for several reasons.

1. Morality needs a Mind
   Theism proposes an ultimate mind in God; Atheism proposes that matter is ultimate, without God, and human minds if they exist at all (contra, eliminative materialists) are merely features of brains. Consciousness is glaring problem for atheists leaving the most direct and obvious realm of the human experience, our own first-person experience of it, a crushing conundrum for atheistic materialists. Yet everywhere we encounter morality we find it to be a mental thing, some set of judgment, knowledge, cognition or intuitions concluding that, for example, Incestuous rape, assault, and murder are wrong. As such, theism offers an ultimate mental basis for morality while atheism struggles to justify the existence of mind with only non-mental causes behind it
2. Morality needs a Lawgiver
   Theism proposes an ultimate lawgiver for prescriptive (moral) laws; Atheism has only descriptive laws, never prescriptive ones. The laws of nature are descriptive, not prescriptive. Now we can reliably use them as if they are prescriptive, but it’s been passe since at least the start of the 20^th century to think of natures laws as immutable prescriptions—like Relativity theory and Quantum theory never happened. Likewise with morality, we may concoct legal codes and regulatory laws—but that does not get us objective moral facts but instead legal conventions, a kind of relative morality. Yet, groups can and have been wrong—such as antebellum slavery in the U.S.
3. Morality needs Truthmakers
   If you state a moral claim like “That girl should not have been brutalized like that” that statement could be any number of things. It could be a feeling, a desire (against something), or even gibberish. But these options, at best, would get us only relativism and there are many problems with relativism that make it a bad choice for serious ethicists. If that claim is going to be true, however, it needs a truthmaker. People can say what they want, but their statements aren’t true unless they correctly represent reality. In this way atheists struggle, since nature has no “oughts.” No claim about what “should or shouldn’t be” can ever correctly portray reality of nature has no “should” or “shouldn’ts.” Nature just does stuff or it doesn’t. It doesn’t deal in subjunctive moods of “if,” or “let’s suppose,” or “should/shouldn’t.” This problem is also known as the “is-ought” problem or the “naturalistic fallacy.”
4. Morality needs Teleology
   Teleology is a fancy term meaning “goal-directedness.” In morality teleology refers to those outcomes for which we should aim, such as happiness, joy, virtue, well-being, etc. But teleology is also treated with hostility in much of the scientific world. It’s considered a relic of the pre-scientific past, a vestige of theology that has since been abolished from serious conversations ever since Darwin showed us that mindless natural forces suffice to explain away all the appearance of goal-directedness in nature. Remember, Darwin, and most evolutionists since him, understand people to be mere animals and in the animal world, the whole appearance of goal-directed designs, he thought, was explained away as mindless, pitiless, amoral natural pressures. Teleology was barred from the biology; yet biology supposedly dictates everything about us. If this account were true, morality collapses since there is no morality, as we normally understand the term, unless there are goals for which we should aim. If the atheist ethicist refuses to let go of teleology, he would do well to run it by his fellow biologists first and see if science accommodates theological relics. Meanwhile, theism has no problem with teleology. Man is mad by God for God-ordained purposes, to be fruitful and multiply and fill the earth, to have dominion and rule over the earth, to love our neighbors as ourselves, to love our enemies and pray for our persecutors, to turn the other cheek, etc. etc.

These four requirements are all difficult if not impossible for naturalistic atheists to produce, yet theism works just fine with all of them, even predicts them by way of a personal, intelligent, creator God.