First, it must be noted that not all religious people suffering terminal illness find it easier to cope with approaching death than atheists do. A generalization of this sort, without statistical analysis to substantiate it, can skew any serious consideration of facts, which then skews any deliberations on social actions or personal decisions.
Having said this, a 2010 study conducted by Professor Clive Seale of England's Barts and The London School of Medicine and Dentistry and published in the Journal of Medical Ethics establishes a strong link between doctors' religious beliefs and their choices to make decisions that may "hasten" a patient's death. Non-religious doctors were "twice as likely" as religious doctors to make decisions that could hasten the approach of death in terminally ill patients (study respondents who were religious reported Muslim, Buddhist, Jewish and Christian religious beliefs).
Seale's conclusion, based on respondents' accounts of patient deaths and on analysis of data, is that religious belief or non-belief is a determining factor in how doctors view death and dying. Those who do believe in God tend not to interfere by hastening a patient's approaching death while those who do not believe tend to interfere by hastening approaching death. This can be extrapolated and tentatively applied to the views of patients themselves.
If the correlation between doctors' beliefs and decisions suggests a correlation between a patient's beliefs and decisions, and if "coping" is defined as making productive decisions in life while under the shadow of death (versus non-productive decisions, while defining "non-productive" as the cessation of life choices and life actions; choices for death and the end of productivity), then it might be suggested that one reason terminal religious patients cope better (cope: decide more productively) is that, from a prohibitive slant, belief in God's laws is a deterrent to actively ending life and that, from a positive slant, belief in the sacredness of life and relationships is an impetus to the continuation of a productive presence.
Another suggested reason, from a spiritual slant, is that belief in the primacy of spiritual reality over material reality (or the primacy of the spiritual in creating the material as quantum physicist Amit Goswami in Quantum Activist explains it) palliates [palliate: to lessen the severity of (pain, disease, etc) without curing or removing; to alleviate; to mitigate (Collins Dictionary)] the patient's pain and suffering making these material realities secondary to spiritual realities relating to worth and relationship, realities that have primary importance.
A suggested correlated reason that terminal religious patients may cope better with approaching death is that they believe they have greater expectation of what lies ahead of them in the next hour, in the next day, in the next life as religious belief most often includes belief about a subsequent life in one form or another (e.g. resurrection, reincarnation, ascension to a higher level etc). Death and dying research, as presented by New Zealand's The Life Resources Charitable Trust, indicates that expectation or lack of expectation plays a significant role in coping or not coping.
Many people are scared of death and dying and do not know what to expect. It is a really good idea to talk to professionals, either medical or spiritual, to gain a better understanding about what to expect. (Life.org, New Zealand)