The Placebo effect: (noun) "A beneficial effect produced by a placebo drug or treatment, which cannot be attributed to the properties of the placebo itself, and must therefore be due to the patient's belief in that treatment."
The reverse is also interesting: psychosomatic illnesses. Though if the main mechanism involved is stress, then this would likewise explain the otherwise unexplainable onset of those issues. Stress wreaks havoc on so many systems in the body.
It does seem to me however that our state of mind influences the hormones released throughout our body. Whether they be stress hormones, or more beneficial ones. And while there are indeed many ways to influence one's state of mind, faith is a very effective method to generate positive states of mind. Similar to how the placebo effect appears limited to affecting conditions originating in the mind (pain levels, nausea, stress, psychosomatic eczema, etc.), faith also appears to primarily affect the mind. Arguably the most controversial aspect of spirituality, faith is by no means the bulk of it. In my estimation, the primary function of spirituality in general, is to give rise to happiness and inner peace in the practitioner. The effects of such happiness can then be observed by reductions in stress, and the associated illnesses that are exacerbated by stress. Also, with increased clarity of mind (from less stress), comes better decision making, among many other benefits.
Within each spiritual tradition / religion, there are people who practice it well, people who totally misinterpret it to the detriment of others, and people who distort it to fulfill their own selfish agendas. Science seems to have perhaps less extreme cases of people using it for nefarious purposes, but has many examples throughout history of people misunderstanding it, and harming others from the conclusions they come to. Many if not all of the criticisms of faith, can likewise be leveled at those who misunderstand and/or misuse science, although I concede perhaps to a far lesser extent globally, and throughout history as well. People in the past misused and abused religion in terrifying ways, but this does not negate the benefits that spiritual practices do provide; just as unskilled and uninformed science mishaps, or possible nuclear wars, do not discredit the benefits that science can provide.
I don't think there has to be a conflict between science and religion. Both are valid for different purposes. Science is for understanding the physical world. Spirituality is for understanding how to be happy. (Other valid purposes exist too, and there is overlap. I oversimplify in order to juxtapose.)