It is interesting how often and freely we use these three elements of thought processing and presume that what we are expressing is being legitimately represented.  Beliefs are readily interpreted as knowledge, and knowledge is often characterized as being true to lend it weight.

However, for the purposes of this discussion let's consider some definitions for these terms so that we can distinguish how these elements are actually used.

A belief is a system of thought that is compromised of the information we have accumulated and stored in our brains.  Collectively this provides a worldview and mechanism by which we interpret new information and assess how our experience in the world should be managed.  What is important to understand is that such a belief does not have any intrinsic validity beyond the fact that it is the way in which data has been organized within our brains and it appears to provide us, individually, with a model against which we interpret the world around us.

Knowledge is going to be more narrowly defined as that information for which we have either direct experience and/or data to confirm that it represents a, more or less, accurate interpretation of the world around us.  Therefore knowledge is always going to have severe limits and the bulk of the knowledge we possess will actually be a product of second-hand information we gain from other sources (trusted or otherwise).

Truth simply represents the opposite of deception.  Although it is often used as a more emphatic way of expressing what we consider to be a "fact", it is irrelevant in that context beyond establishing that the information being presented or interpreted is not the product of deceit.

Other terms that we encounter will involve, "facts", "assumptions", as well as various means by which we extrapolate conclusions based on existing evidence.

So, how does this all fit together?

In the first place we must recognize that everything we think as human beings is dependent on having a coherent belief system¹ and a functional data organization method within our brains.  Some people may balk at this idea of such a pervasive belief system, but we are forced to accept this definition since it is clearly impossible to have verifiable information about everything that we encounter in the world.  Therefore one of the functions of a belief system is to allow an individual to operate in the world as if they actually possessed complete "knowledge" about it.

Knowledge therefore, according to my definition, is the means by which "facts" can be gathered to reinforce, or refute, different aspects of the information contained within our belief system.  Often, it is asserted that something is "true" or "factual", but neither of these are actually relevant since the only consequence of knowledge that we are interested in is accuracy.

However, even in the interest of accuracy we are often forced to generalize because a "fact" simply isn't accurate as a general statement or description.  Asserting that a particular "fact" is true is simply a mechanism we use to argue that we are presenting it as being free of deception.  Factual information is, by definition, true.  What should be questioned is its accuracy or applicability to any particular circumstance.

As it turns out the fundamental problem is in interpreting the knowledge we possess against the belief system we hold.  If we believe that the world is subject to being understood by query, then we will tend to hold a more scientific view of things.  If we believe that the world is full of mysteries that can never be understood, then we will tend to be more inclined to accept that external agencies are at work.  I want to be clear that I'm not arguing that religion and science are incompatible with one another, because there are clearly many people that can comfortably bridge that gap within their respective belief systems.  The difficulty comes from the extrapolation of knowledge into unknown areas.

Invariably this leads us to another area of human thought, which is represented by logical systems that we have invented that are an attempt to remove or control the intrinsic bias we possess based on our belief systems.  Mathematics and symbolic logic are mechanisms that we can employ which force us to follow specific rules in how information can be manipulated.  By rigorously emphasizing such rules, we have a means by which we can manipulate symbols representing particular pieces of information and arrive at conclusions that are consistent within such a logical system.  As a result, we tend to view these disciplines as being indicative of "truth" or representing reality.  

Similarly the scientific method is a logical means by which queries are to be formed and answers established based on various criteria intended to avoid bias and increase accuracy.  In virtually all cases, the underlying difficulty occurs because our own bias tends to support our desire to make information fit within our existing belief system.

While many people may lay claim to objectivity and exemption from the influence of a belief system, such a situation simply isn't possible.  Besides being a vehicle by which we must operate in the world, a belief system is also the mechanism whereby we interpret new information as being erroneous or acceptable.  It is easy to argue that one simply has to "follow the evidence", but the "evidence" is subject to the interpretation of our belief system, so, once again, the problem comes from interpretation.  It is simply not possible to maintain an "open mind" about every piece of data we encounter, so invariably we must pick and choose how information is to be processed and that filter is our belief system.

As an example, consider the relatively simple case of global climate change and the melting of the Arctic ice.  Presumably it should be a trivial matter to gather basic data consisting of the amount of ice at any point in time and the average temperature.  However, even with this data we find that the "facts" are subject to enormous variations in interpretation based on individual belief systems.  To some, this is clear evidence of human effects on the ecosystem, to others it isn't clear that humans are responsible, while to others they may dispute that the ice is melting at all, or even if true, that it is a bad thing.

This does not lead us to some sort of intellectual relativism where every belief system must be considered a valid representation of the world, but it is something that we must come to terms with if we are to understand why we behave as we do.  This has also lead some to propose that such knowledge-based interpretations and decisions should only be made by "experts" that are truly qualified to understand the nuances of their area of expertise.  However, why would we believe that such expertise would lead to better decisions²?  

The issue from all of this is that we shouldn't confuse our own beliefs with knowledge, regardless of how well it appears to work for us.  The biggest bulk of the information that we rely on for our daily lives can simply never be verified (or scrutinized for accuracy), so while such an arrangement may work well for us, we should always be cautious in presuming that what we believe is synonymous with what we know.

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¹ Note that belief is NOT synonymous with faith. Belief is tempered by our direct experiences in the world, so that even outlandish ideas may gain acceptance if they appear to "work".  Faith is simply accepting pieces of information despite there being no evidence for their existence.  Any information that is capable of verification, regardless of how faulty or tenuous it may be, cannot be said to be faith.  In effect, faith can be considered the axioms of one's belief, so it is not a uniquely religious phenomenon although that is how it is commonly considered.  

² Most difficulties in conflicting belief systems has little to do with knowledge or belief, but is driven by politics.  Therefore to delegate decision making power to "experts" is suspect because there is no reason to credit them with being more or less politically neutral than any other individual regardless of how well, or poorly, informed they may be.