What is the difference between cum, quia and quod in a causal clause?

This is only a partial answer and in need of further substantiation, but I'll post it for what's it's worth.

First, as to the difference between quod and quia. Though these are often treated as interchangeable, and both can take either the indicative or the subjunctive depending on whether the reason stated is the writer's or is ascribed to someone else, there may be a difference in usage. Allen and Greenough 540 state:

Note 1— Quod introduces either a fact or a statement, and accordingly takes either the Indicative or the Subjunctive. Quia regularly introduces a fact; hence it rarely takes the Subjunctive. Quoniam, inasmuch as, since, when now, now that, has reference to motives, excuses, justifications, and the like and takes the Indicative.

That is, it seems that quia and quoniam rarely express a reason ascribed to someone else, but normally state a reason that the writer vouches for, while quod has no such restriction. That said, it would be good to see other corroboration or a more detailed discussion of this difference. Woodcock's New Latin Syntax does not, as far as I can find, mention it; I wouldn't be surprised if the Oxford Latin Syntax has something to say about the question (the table of contents suggests it might), but I don't have access to it at the moment.

As to causal cum clauses. My comment which prompted your question wasn't meant to suggest that these are clearly distinct from quod or quia clauses, only that they shouldn't be assumed a priori to be synonymous. "Cause" is a notoriously non-unitary concept, and true synonymy in language is rare. So-called "causal" cum clauses are not really a clearly distinguishable subset of subjunctive cum clauses generally, which are normally about describing the circumstances under which something happened; thus Woodcock (p. 192): "Both the causal and concessive (adversative) uses of cum are really branches of the generic or descriptive use". The difference between a circumstance with explanatory value and a cause in a stricter or more logical sense is not easy to define, but possibly some such distinction can be drawn between cum clauses and quod/quia clauses. In terms of English equivalents, a translation like "seeing as" (as opposed to "because") can express the former and is often suitable for cum clauses, less so for quod/quia clauses.

The best answer that I can give is to adapt the notes I made years ago during instruction in prose composition [which I am doing by inserting comments in square brackets].

ADVERBIAL CLAUSES — CAUSAL

The word cum has a whole list of meanings, as conjunction, preposition and adverb. The form quum is often found instead of cum, and these two are more or less interchangeable as causal particles — that is, to introduce causal clauses — both being found throughout classical texts. The form seen is often that preferred by the redactor [which may be an obstacle when a corpus is consulted to deduce a trend, or a rule based on frequency].

1. [A little contrary to what the question says] The verb following cum is normally in the subjunctive : but there are a few exceptions — the indicative can be used when the action of the verb is future or present (e.g. after a few 'verbs of emotion' such as laudo, doleo, gaudeo). Note that in adverbial clauses that are NOT causal but temporal, introduced by cum = 'when', the indicative is legitimately used in various situations.

2. [For each of the words in question] English from Latin is chosen to suit the style of the translator: since, as, because, seeing that, for the reason that etc. For doing into Latin, on the other hand, Quia and quod can, like cum, be used to translate any of these and, like cum, are used with the subjunctive — with indicative ONLY if the reason is given with certainty:

advenerunt quod cibum susciperent but advenerunt quod cibo egebant

3. Since causal clauses are syntactically adverbial, the introductory particle can refer to a reason shown in the main clause:

cibo egebant, idcirco advenerunt quia cibum susciperent

4.To conclude : When used as logical conjunctions, the differences between cum, quod and quia are minimal. The latter is most often used by the subject of the principal clause in expressing an opinion and is therefore considered rather more emphatic : such usage is not obligatory. The differences hardly show in a good, idiomatic translation from Latin to English : such difficulty as occurs lies in the opposite case, of choosing the most suitable word of Latin to suit an English original. Here, too, teasing out the niceties of difference is a somewhat arid pursuit, for there may be many ways of saying the same thing in English on these occasions, just as in Latin : in the end, it comes down to personal style and preference. In that connection we should remember that not only clarity, but also elegance is desirable in a Latin prose that aspires to high quality.