The philosophy underlying the sciences is known as "logical positivism", and was first codified around the turn of the 20th century. At its core is a fairly simple, obvious epistemology that permits metaphysical and ethical questions to be ignored entirely, freeing the sciences to concentrate on the questions the scientific method can actually work on.
The basics of positivist epistemology are simple: All statements are either positive or normative (we'll ignore for now the possibility of completely nonsensical statements). A positive statement is a statement about the empirically measurable properties of one or more physical entities. For example, "This camel has three humps," is a positive statement, because the camel is a physical entity, and its number of humps is an empirically measurable property. "No camels have three humps," is also a positive statement; it speaks about the same empirically measurable property of a large number of different physical entities, the camels.
A positive statement can be either true or false, and indeed all positive statements are one of the two (though we can't always know which). For example, though we can never know how many times George Washington blinked on April 26, 1780, there is a correct answer to the question. George Washington is a physical entity, and the number of times he blinked is empirically measurable. Each of his blinks had an impact on the world around him -- for example, by creating tiny air currents -- and in theory, given infinitely accurate measuring devices, unlimited computing power and speed, and the right theoretical models, we could reconstruct his blinks on that day.
Sometimes a statement can be defined as positive not because it is itself about the empirically measurable properties about a physical object, but rather because it has logically necessary consequences which are positive statements. For example, the claim that a physical object exists is a positive claim because, although existence itself is not an empirically measurable property, in order for a physical object to exist it must have certain properties. For example, Russell's teapot is a hypothetical that claims that there exists a small china teapot suspended exactly opposite from the Earth in the same orbit, so that the Sun is always between us and it. If the claim is true, then the teapot, as a physical object, must have mass; this is an empirically measurable property. Thus, Russell's teapot is a positive claim.
Claims which are not positive are normative. An example of a normative claim is, "Eating kittens is wrong." Eating a kitten is a physical event, but "wrongness" is not an empirically measurable property. Normative statements are too dependent on the speaker to have a truth value; I might find eating kittens to be wrong, but you might not, despite no change in the physical event.
Any normative statement can be turned into a positive statement about Bob by adding "Bob says that..." to it. (Substitute the speaker of your choice for Bob.) For example, "Eating kittens is wrong," is a normative statement about eating kittens. "Bob says that eating kittens is wrong," is a positive statement about Bob. We can take empirical measurements (for example, asking Bob and listening to his response) to determine if this claim about Bob is true.
Some people argue that experiences such as "wrongness" or "the numinous" or "freedom" are empirical phenomena because they correspond to particular states of the brain. While probably true (certainly there is no evidence to suggest that any state of mind is anything other than a state of the brain), this is really only a variant of adding "Bob says that..." "This brain responds to kitten-eating by firing the wrongness neurons," is a statement about a particular brain; it still does not point to any inherent properties of kitten-eating.
As I mentioned above, normative statements cannot be regarded as "true" because they depend too much on the speaker. They qualitatively differ from untestable positive claims like George Washington's blinking: Even given unlimited measuring equipment and infinite computational power, I cannot detect a physical property of wrongness or justice or value because there is no such thing. The most I can accomplish is some version of "Bob says that..." There is no correct answer to the kitten-eating question, only opinions.
If no normative statement can be true, then no normative statement can be false. If a normative statement were false, then its negation would be true. For example, if we declare "Eating kittens is wrong," to be false, then we're saying "Eating kittens is not wrong," is true. But that statement is also a normative statement, and therefore cannot be true. Thus, we must conclude that normative statements have no truth value.
The founders of logical positivism went further, and declared normative statements to be meaningless. From a purely scientific perspective, that's true: science cannot work with or generate normative statements, and it's dangerous to try. From any other perspective, however, that's nonsense. Linguistically, culturally, neurologically, normative statements absolutely do have meaning -- I'd like to see someone try to get through a day without acting as if value is a meaningful concept! -- and so, except when we are considering scientific questions, it would be foolish to dismiss normative statements just for being normative.
Let's put this another way. A statement is true if it accurately models our physical reality. It is false if it contradicts physical reality. All true statements and all false statements together make up the set of positive statements; hence, all positive statements are attempts to model physical reality and vice versa. A statement which does not model physical reality is normative; since it neither accurately models nor contradicts physical reality, it is neither true nor false.
Claims of existence are a somewhat tricky case. The problem is that most people are not philosophically rigorous, and hence language is not, either. So, for example, when Russell claims his teapot exists, he is claiming that it physically exists and therefore has physical properties, such as mass and position. When someone claims that love exists, however, they are generally not saying that there is a substance called "love" that physically exists and can be detected; rather, they are saying that a particular pattern of behavior occurs in human relations. This is still a positive claim -- human behavior can be empirically observed and recorded, and we can see if it's consistent with the love hypothesis or not.
However, when someone claims that free will exists, they are using a different definition still. They are not (generally speaking) claiming that there is a "free will" substance with physical properties. Nor is the claim that free will exists something that can be tested by observing human behavior; human behavior is exactly the same whether it is caused by deterministic, but highly complex and impossible to predict, natural phenomena or free choices that "hide" in the "highly complex and impossible to predict" part. There are, in fact, no statements about physical phenomena that follow logically from "Free will exists"; thus, it is not a positive statement.
Similarly, one could claim that a teapot exists in another universe, entirely distinct from our own. Since there is no attempt here to model physical reality -- we are talking about some other, purely hypothetical, reality -- the claim is entirely normative. Depending on the context in which the claim is made and how seriously we are expected to take it, we might call such a statement fiction, myth, allegory, a thought experiment, a parable, or any of a hundred other related concepts.
So, we come to the big one: "God exists." Is this a positive or normative statement? Well, it mostly depends on what you mean by "God." If you're an animist and believe that the universe has a guiding spirit akin to the human will, that's not a positive claim any more than free will is. If you're a Christian and mean that a first-century rabbi healed sick people, revived the dead, preached, died, came back from the dead, and then left this universe (reducing its total mass-energy by a hundred kilos or so), that's very much a positive claim.
A universe in which Jesus came back from the dead is different from one in which Jesus did not come back from the dead. Given the right equipment, we could measure the universe and determine which occurred. This is not a matter of opinion; it happened or it didn't. I think it didn't, but even if we could prove it didn't, most Christians would adjust just fine. They've already adapted (except for a few vocal nutters) to viewing their creation myth in metaphorical terms, once its positive claims were shown to be false. There's no reason they couldn't do the same with their main hero-cycle myth, as well.
Most religious people, like most people, don't particularly care much about modeling physical reality. Most religious people are just fine to say "Leave it to the scientists to work out how the universe functions; God is why." Just as free will has no impact on human behavior, and thus is not a positive statement, so does the existence of this sort of God have no impact on the behavior of the physical universe. Such a God is not a positive claim. There is no difference between a universe with such a God and a universe without such a God; they are the same universe.
Since they are the same universe, neither the believer nor the atheist can be right or wrong. There are no competing models of physical reality here; only two different ways of looking at the same physical reality. Reason does not compel a position on the question; we are free to believe as we choose.