As tests of truth, both consistency and coherence are relativistic criteria. Instead of measuring a belief directly against an outside reality, they measure a belief against other beliefs. If that belief is consistent with or coheres with already-held beliefs, then it is considered true. If it fails to be consistent or to cohere, then it is discarded as false.
Consistency can be employed in two different ways: loosely or rigorously. Loose consistency simply means that a group of beliefs do not contradict one another logically there is no actual need for them to be related. Thus, the statements snow is white, light travels fast, and today is Tuesday are all consistent under the looser standard.
The problems with such a loose understanding of consistency are pretty evident any or all of those statements could fail to be true when compared to reality but still treated as true merely because they dont contradict one another. Loosely consistent statements wont contradict because they are unrelated, but they are no good guide to truth and falsehood.
Because of this, anyone employing consistency as a test of truth is more likely to focus upon rigorous consistency. According to this criterion, statements are not simply related to one another, but in fact any one statement logically and necessarily follows from the others. This sort of consistency is readily apparent when it comes to systems of mathematical formulas unfortunately, that is about all it is good for.
Rigorous consistency is very limited because the requirement that any one proposition follow from the others means that no new information can ever be produced. Every belief is deduced logically from other beliefs and induction is ignored; yet induction, even though fallible, is how we produce new ideas and eventually new knowledge. Rigorous consistency is also flawed because there is no requirement that it take into account all knowledge and information you can limit such a system to just a few propositions and whatever follows from them.
Coherence is a potentially more productive and interesting test for truth, perhaps explaining why it is more popular than any of the versions of consistency. A coherent system is one in which all facts of experience are integrated, but it does not require that all facts follow deductively from the others. Instead, it merely requires that related facts be positioned appropriately and that nothing ultimately contradicts anything else. Everything coheres because it fits together coherently.
Assuming that the system of beliefs you already rely upon are accurate, then demanding that a new belief cohere with them may be the most reasonable and reliable test for truth one can use but that is a big assumption to make. If your belief system is seriously flawed, then anything new that comes up is likely to be rather unreliable. The key, then, is to start out with a belief system that is reliable as possible.
Of course, we cant object to the coherence test of truth too strongly. To do so would require, at least implicitly, arguing for a system of beliefs that is incoherent and that isnt possible to do consistently. Thus, even objections to the coherence criterion of truth must assume, at least to some degree, the validity and importance of coherence when it comes to beliefs.