The Stoics did not have the benefit of Christian revelation. Stoicism has been called the “porch” of Christianity. Its resignation and detachment take a person to the threshold of Christianity, but don't take him all the way into the house. Although the Stoics taught belief in a monotheistic God, their cosmology excludes a transcendent God. Theirs is (at least in many of its thinkers) a type of monistic materialism, resembling pantheism and similar to Hinduism and Buddhism–and wholly inconsistent with Christian cosmology.
Stoicism also deviates from Christianity with respect to important ethical issues. Suicide, for instance, is acceptable, maybe even required, if given adequate reason--a principle Seneca taught and practiced.
The Stoics were committed to a lopsided sense of justice that has no place for mercy. Their respect for and development of law gave rise to natural law theory, which has informed Christian jurists for nearly two thousand years, but their merciless justice gave rise to troublesome results, like Marcus Aurelius' persecution Christians, which he thought was necessary to preserve order (I hasten to note that his conclusions about Christianity resulted from a misunderstanding of Christianity, which in turn was caused by fanatics at the fringe of the Church).
The Stoics also placed too much emphasis on reason. As mentioned, Stoicism starts with a monism that teaches that a divine reason (logos) pervades the universe, including each person. When the Stoics tell a person to "live according to nature," they are referring to man's nature as a rational being. Humans, the Stoic says, can use their reasoning capacity to know the divine reason and conform their lives to it.
Emotion is anathema to Stoicism. Not only does stirred emotion (passion) distort reason, it conflicts with the Stoic's basic theological precept that divine reason guides and directs the world. Whatever happens, happens, says the Stoic, and there's no use rejoicing or moaning over it; it all ultimately leads to wherever divinity wants it to lead. It is the ultimate form of detachment.
Although Christianity also celebrates reason and counsels dispassion and detachment, such things have their limits. Reason cannot fathom all the workings of God. The Christian also celebrates emotion, within limits. Jesus's resurrection is the Good News, and a reason for cheerfulness and celebration. Christians are charged with spreading the Good News with gladness. Christians love others. Christians hate sin. We are not detached from all things, including the well-being of our soul. We live in the Stoic present, yes, but look to the future with hope, joy, and peace.