What are the fundamental differences between the belief system of science and religious belief systems?

While both deal with answering the "What?" questions; One deals with the answers to "What happened and who did it? the other deals with "What happened and How it happened?"

Religions people generally personify the universe, that is they look for the cause of things as the result of some intentional agency. Science on the other hand looks at the cause of something happening as the result of natural laws distinct and separate from any intentional agent. While they both may agree on "What happened" they differ as to the cause of it happening.

Let's start with the things they have in common on a conceptual basis:

Both Science and Religion have an unfounded a-priori assumption. In Religion this is the belief that the Divine has manifested itself in some discernable way (otherway how would we know about it). In Science this is the assumption that the Scientific Method can be used to make usefull claims about the world.

The second might seem more self-evident, living in this technology filled world, but there is no way to prove that this effect is intrinsic and not just happenstance.

And what do they have in common in practice?

While Science as a concept is fairly different from Religion, in actual practice there are some overlaps. The first and foremost is the concept of Scientism, the "believe in Science". At it's most extreme this means a complete rejection of anything that can't be proven by the scientific method.

As mentioned earlier, there is nothing aside from common human experience that proves that the scientific method is correct. While this is certainly is enough to make a case for following this stance, it is strictly speaking nothing more than a religion.

In addition there is a certain amount of reliance on authority, to overcome the limitations of the human lifespan and lack of resources: While in theory, one could repeat all experiments that are the basis for our scientific knowledge, in practice we have to rely on established institutions, and trust that these institutions have done the work for us. That is to say, that both religion and science rely on authority, but the sources of the trust placed in these authorities differs (or should differ at the very least).

Now where do they differ on a conceptual basis?

The on thing that Science claims for itself that sets it apart from Religion is repeatability. Anyone can repeat any of the experiments upon which our scientific knowledge is based.

One could draw paralels between the inability to reproduce science, and the requirements other Religions place on their believers (e.g. "you can't hear god because you're not pious enough", "god only speaks to the chosen ones"). The difference is, that the factors that prevent people from repeating scientific experiments are not build into Science itself.

And what seperates them in practice?

First and foremost, science has a direct and concrete effect on human lives, even if we ignore the secondary effects caused by belief in science. Religions do claim the same but even if we assume that all reported Miracles are actually true, only a negligible ammount of people seems to be affected.

On average human experience seems to say that science works, while faith doesn't, to the point where most popular religions concentrate on intagible effects (e.g. rewards in the afterlife, faith gives hope, faith is the basis of morality).

There also seems to be a difference in how scientists and religious people approach arguments. While science is much more concerned with the results of it's base assumption, religions seems to have a much bigger focus on proving their base assumption. Again this is because Science proves it's own worth by delivering results, while Religion, due to the lack of concrete (physical) results, has to prove that it comes from a solid foundation.

To summarize: Science doesn't have to be true, it only has to work such that everyone can see it's usefullness. Religion has to be true because it's usefullness isn't readily apparent.

The most fundamental difference is that religions believe that the mind (in its wide meaning) is the source of creation, the source of all phenomenons, while science believes that their is no mind and that matter itself is the source of everything we see. This is why both dogmas have always been in opposition.

While you can stumble upon genuine scientists who seek the truth, the scientific method does not prevent science and scientists from following an ideology that originates from the Enlightenment (century of lights), as shows the persistent paradigms in every fields. Indeed, pick any officially accepted theory and you will notice that it explains a phenomenon by matter being the source and systematically rejects a possible implication of the mind.

For details see the work of Rupert Sheldrake https://www.youtube.com/watch?v=JKHUaNAxsTg

While most religious belief systems affirm the existence of certain things, the belief system of (natural) science tends to deny the existence of certain "not-reproducible" things.

I don't think this is true. Science - in the limited scope you meant here - only deals with things that can be demonstrated by reproducible experiments. It can't deal with what is, by design or nature, not subject to such requirements. Are there invisible fairies in my garden? We cannot know, because the claim is unverifiable by design: the fairies are invisible, and so we cannot see them. Can we communicate with the dead, if we have faith? We can't know, bucause the claim is unverifiable by nature: if we were to make an experiment, we would already lack the necessary faith.

It doesn't follow that those things don't exist. It follows that science cannot reach to conclusions regarding them. Collaterally, there is a "belief", which can not experimented in a "scientific", reproductible way, that things that have inbuilt unverificability clauses, such as invisible fairies or any phenomenon dependent on faith, are either not important, or so rare that they cannot be accounted for in any meaningful way.

On the other hand, scientists seem to believe in the Big Bang, the existence of dinossaurs, or dark matter, all of which are unsusceptible of repeatable experimentation. So, while repeatable experimentation is certainly an important part of science, it cannot be all that is in science.

Let's look at two examples to make this question more concrete. There are many examples in fiction of interactions of no longer living persons with living persons. But suppose this would happen to you in "real life". Wouldn't you try to explain it away (or at least keep it for yourself)? Perhaps you manage to settle for a "I don't know" position.

There are many real life accounts of interaction with the dead. But, in short, the problem is that either whatever information the dead bring to us can be checked by the living, or it cannot be checked. In the first case, the dead can usually be shaved off by Ockham's razor - they are unnecessary to explain how such information was obtained. In the second case, we cannot know if such information is actual information, or just a fantasy. The dead could, of course, clear up the issue by informing me the numbers of the incoming lotto - but then it seems they have a strict (and convenient) moral code that forbids them from doing that. And when they otherwise talk about the future, they tend to adopt a contorted (Nostradamic) style that makes it impossible to understand what they are predicting.

Now let's look at the different belief systems of (natural) science and religion. Is there any need to fit this into a scientific belief system? Isn't (natural) science concerned with "reproducible" things, so that science doesn't even need to bother whether certain "non-reproducible" things are "real" or not?

It cannot be so indifferent to such "non-reproducible" things, as they will contaminate the reasoning and result in non-verifiable hypotheses. And so, they must be purged.

Can the same position also be used for religious belief systems? Probably not, because this touches the kind of questions that religious belief systems are concerned with. But what about the opposite case, for example heliocentricism? That's a scientific theory (or fact) after all, so it shouldn't worry religious belief systems too much. But apparently it did.

It did, but that due to certain particularities of the Abrahamic faiths, which imply the inerrancy (at a least to some level) of given, dated, sacred texts - sacred texts that make several assertions about existing or past states of the universe. It is worse for literalists, such as a few fundamentalist Protestant sects, and less of a problem for non-literalists, such as the Catholic ortodoxy, or rabinic Judaism, that can always fall back into an alegorical interpretation (yes, the Bible says God created the world in six days, but who says that a billion years is not just a day for God?)

From Philosopher3's answer:

Religion has a conclusion (i.e. deity, after life, creation, etc.) and then finds evidence to support those conclusions.

[...]

Science on the other hand has evidence (i.e. an object falls at a contestant rate regardless of weight) and then draws a conclusion from it.

It is not like that. Science starts with conclusions (theories) and only then seeks for evidence. The difference is, science seeks for evidence that contradicts its "conclusions" (ie, theories), and only sticks to conclusions that resist such trial. Religion does not seek evidence at all, and tends to reject (or misinterpret) evidence that contradicts its conclusions.

While most religious belief systems affirm the existence of certain things, the belief system of (natural) science tends to deny the existence of certain "not-reproducible" things.

Why would the mere fact that you cannot reproduce something lead you deny a thing existence? I cannot reproduce a Boeing 747. Does this now somehow call into question the existence of such a thing? Nope!

Let's look at two examples to make this question more concrete. There are many examples in fiction of interactions of no longer living persons with living persons. But suppose this would happen to you in "real life". Wouldn't you try to explain it away (or at least keep it for yourself)? Perhaps you manage to settle for a "I don't know" position.

Why would you think they did not doubt it Thomas certainly did doubt it. I do not think it was harder for any Apsotle to believe it than what it would be for any humans today.

But how clear is the distinction between the two really? It is clear that the domain of expertise of a priest is quite different from the domain of expertise of a scientist. But does that really stop the corresponding belief system from making assertions about things outside of this domain of expertise? (And what is the position of philosophy here? It seems to have an opinion on both science and religion. But is it more part of (natural) science, or more some sort of "impartial" observer?)

Now this to me is a very interesting question. It was a question that plagued a certain person called Stephen Jay Gould and he got famous in academic circles for the answer he proposed to this very question.

He posited that both science and religion have their own valid teaching domain or authority. As long as they are both within the bounds of their own "teaching domains" then their should be no conflict between the two. He coined the term Non-overlapping Magisteria or NOMA in reference to this.

Now personally I hold to POMA or partially overlapping magisteria. Their are some discoveries in science that hold great discussion in religious circles. If you take the discovery of background radiation for instance. This in essence began the road to a finite universe which was remarkable and religiously relevant discovery. It also sparked a resurgence in Aquinian thought which was for centuries held to be disproved by physicist.

Their are some who even hold to a completely overlapping magisteria (Coma) like Richard Dawkins for example. He believes erroneously (to me at least) that the God hypothesis is a scientific hypothesis (As he explains in his books) and you should use the scientific method to ascertain it's validity.

You can read more about it here.

http://en.wikipedia.org/wiki/Non-overlapping_magisteria

The key differences are as follows: Religion has a conclusion (i.e. deity, after life, creation, etc.) and then finds evidence to support those conclusions. This is most evident by the theological fallacy "God is in the Gaps".

Science on the other hand has evidence (i.e. an object falls at a contestant rate regardless of weight) and then draws a conclusion from it: Newton's theory of gravity, which was later changed by Einstein and his general theory of relativity.

Science is a field of study where statement of theories are made, in which there are probabilities or possibilities. These statements can be proven or disproven on the basis of controlled experiments.

Religion is a field of beliefs in which statements of certainties are made. These statements are typically not questionable, because they are not subject to proofs or disproofs. Depending on ones inclination and faith, one may only argue that any statement within religion is inherently true, or untrue.

Fallacies are beliefs in theories that are not questionable due to how the theory is defined.

"Science has proof without any certainty. Creationists have certainty without any proof."

This half-joking quote from Ashley Montague highlights for me the critical difference between religious belief and scientific belief: the beliefs of the religious are effectively unquestionable, whereas scientific beliefs can generally only be accepted as legitimate if they are questionable. In other words, at some core level religious beliefs seem to require a "leap of faith", belief without logical reason or justification, whereas all "beliefs" or theories of science can only be said to be valid if they are—on some level—questionable (i.e. falsifiable). Scientific beliefs are never held in any light such that they become beyond reproach, and the scientific method is all about testing and retesting theories, modifying our understanding of the world around us. In contrast, the religious doctrines of the major world religions are largely fixed, the sacred texts are static and unchanging — at best, believer's views and interpretations of the doctrines have changed over time rather than the actual texts themselves.