There are two separate issues here. Some religious philosophers and theologians distinguish between being and existence because for God, who is the source of all existence, and therefore precedes it logically, can not be meaningfully said to exist. The same point can be phrased in reverse, to wit God exists, but does not have being. As Plotinus puts it "The One is all things and yet no one of them. It is the source of all things, not itself all things, but their transcendent Principle... So that Being may exist the One is not Being, but the begetter of Being."
For the debate you are referring to the more common terminology is to distinguish not between "existence" and "being", but between "existence" and "subsistence". The main motivation comes from the necessity to reason about non-existent objects, because for example we do not know that they do not exist from the outset, or because we want to prove that they do not, and therefore can not assume it, as e.g. in proofs by contradiction. The puzzle is dated back to Plato who reasoned (in Quine's rephrasing) "nonbeing must in some sense be, otherwise what is it that there is not?". Here is an elaboration by SEP:
"In order to assert “there are nonexistent objects” without implying nonexistent objects exist”, one has to suppose that sentences of the form “There are Fs” mean something different from sentences of the form “Fs exist”. Some philosophers reject a distinction between “there is” and “exists” (see, for instance, Lewis 1990, Priest 2005, Quine 1953), some philosophers (e.g., Meinong 1960, Parsons 1980, Zalta 1988) think that there are good reasons for making this distinction. Some of the latter think that the distinction between “there is” and “exists” is rooted in ordinary language, but others deny this firmly (see for instance, Geach 1971)."
On Meinong's view subsistence is the "under"-existence that allows us to talk and reason about non-existent, even self-contradictory (round squares), objects. Quine's solution is to avoid such extravagant objects by applying paraphrase, namely by rephrasing sentences that seemingly involve them into ones that do not, but have the "same" meaning. This uses so-called definite descriptions of Russell. E.g. "the current king of France is bald" is rephrased as "there is someone who is currently the king of France and who is bald", which only requires a variable and two predicates, no objects.
Russell's solution is elegant but it is disputed that paraphrases by definite descriptions preserve the original meaning. The issue is tangled up with the old problem of "intentionality", a seemingly universal directedness of our mental states at objects pointed out already by medieval scholastics, and promoted by Brentano in 19th century. It seems that our thoughts about dragons are directed at dragons as intended objects, without paraphrasing themselves into predicate calculus. Quine was interested in transforming natural language into a "cleaned up" scientific language, so to him this is just as well, but if one wishes to stay closer to the semantics of natural language Meinong's approach to intentional objects as subsistent might be more germane.
In a sense, the debate is about linguistic preferences, but it is not as wanton as it seems. Developing a faithful semantics of natural languages (how they relate to their referents, if any) is considered a notoriously difficult problem, but at least partial solutions to it are needed in developmental linguistics, psychology, AI research, etc., to answer questions like how infants connect words to their referents, or to build machines that can do the same without supervision. It can be seen as part of the debate on how best to develop such semantics. For more on this see Is the use of inconsistent definitions a logical fallacy? and Nonexistent Objects.