Religion & politics, aside from being the two topics you are never supposed to discuss in polite company, are also considered by modern standards to be two realms that should never mix. That is to say, your religion should never influence your politics and (to a lesser extent) vice versa. However this view does not seem to hold up neither in theory nor in practice except by adopting either a very limited and superficial understanding of religion or a position of a purely economic role for the State (and in this latter case, not even so).
To start in the general, we must focus on what both religion & politics are. Religion, although many people have many definitions for it, I believe it may be concisely defined as rendering unto a greater being their due. One may ask what is inherently due to such a being, but the answer is that evidently humanity has always rendered homage to that which it considers to be greatest, whether this be pagan deities, God, or more recently with one's own identity. In one manner or another there is some entity which we consider to be worthy of praise, worship, and respect to a greater degree than any other. This, in turn, will generally come accompanied by a philosophy which explains why this entity ought to be praised, worshipped, and respected above all other beings, and evidently such a philosophy will also end up having repercussions on how, then, we ought to view other beings in accordance to how each being relates ultimately to this greatest one.
Politics, on the other hand, is the art of public policy; defining and enforcing the rules by which we live together in a society. Some of these policies are defined somewhat arbitrarily, in that it is simply used for the convenience of having a common standard. An example would be the case of a policy describing which side of the road to drive on (the left or the right), as it ultimately does not matter so long as we all abide it to avoid collision. These we can refer to as conventions. But other policies are defined not by arbitrary or simply pragmatic judgement, but rather by moral or ethical imperative such that the absence of such a policy or of its enforcement would be permission to injustice. An example of this would be the prohibition of murder, without which the injustice of murder would go unpunished. This we can refer to as law. It is the latter of these two concepts which will be important for this discussion. Furthermore, law, because it is not something arbitrary, will necessarily be based on some value or principle which in turn proceeds from some philosophy or worldview.
When considering these two definitions, although there may certainly be manners in which one should not affect the other, it should be evident that there are also a great many ways in which these two coincide and influence one another, and more importantly how religion influences one's politics. For if one simply considers religion to be a set of customary cultural rituals with no further significance other than perhaps the sentimental then certainly such religion (if we can call it that) will hardly influence one's politics. But true religion which is defined by adherence to something greater and additionally provides us with an understanding of the world and how entities relate and ought to relate to one another, such religion will inevitably provide us with values, values which we are obliged to bring to the political sphere since it is not merely a matter of customs but of truth and of justice.
For this reason it is absurd for the modern secularist to claim that we ought not to vote or enact policy in accordance to our religion, for then in accordance to what are we supposed to vote and enact policy? It would be one thing if we were speaking of certain ceremonial customs which it would not make sense to impose upon those pertaining to another religion: precisely those superficial elements of any religion which comprise its outward manifestations. However, in regards to those matters which are not merely superficial manifestations of piety, but substantial matters of ethics it is indeed unjustly discriminatory and undemocratic to say that they ought not to be enacted as policy on the basis of being proposed by a religious people. For at the core of this argument lies either a tactical attempt to invalidate the religious person's opinion (conveniently only applied when said religious person disagrees with them) or an ignorant view that all ethical values of the religious person (or at least those where disagreement is found) are merely the product of arbitrary and groundless superstition. In either case the objective is to avoid having to argue over the reasons for the disagreement: because the position is influenced by the proponent's religious beliefs it therefore ipso facto is not to be considered or in any way be taken seriously.
The fear, of course, when speaking of this subject is that certain moral values that one does not agree with shall be imposed upon the whole of society, even if one does not agree with said beliefs. Yet although this fear can sometimes be reasonable there remains the fact that all law is ultimately an imposition of moral values on society, and a there will always be a segment of society that does not agree with those laws or believes their own breaking of the laws to be the exception. The question with law and moral values is not whether everyone can agree to them but whether they are true and reasonable. Therefore it is to be expected that all persons in a society will advocate policy in accordance to their values and we must accept or reject them on the basis of truth. It is to be expected that vegans will encourage policy which restricts or even bans our access to animal products; I would disagree with such policy not because it is an imposition of belief, but because the vegan attribution of personhood to animals is wrong. Similarly as a Catholic I will encourage policy which seeks to restrict or even ban abortion, not out of some fideistic imposition of values, but because of the truth that abortion is murder. And therefore, just as I would argue with the vegan's worldview to ensure that I have access to animal product, so too would I expect a similar discussion to be had on the issue of abortion.