What specifically is the difference between a 'content clause' and a 'complement clause'?

Here a complement clause is defined as:

a notional sentence or predication that is an argument of a predicate

Here a content clause is defined as:

a subordinate clause that provides content implied, or commented upon, by its main clause.

These seem to be vague definitions that can be interchanged with one other.

This article even suggests that linguists can't agree upon what constitutes a content clause versus a complement.

"There's an arrogance in the scientific community that they know better than the average American."

But we also don't call them finite complement clauses, though many linguists would. The reason is that content clauses are often complements, but not always.

What exactly is preventing the above that-clause from being a complement clause, if we take the definition that I posted to be true?

A complement is a dependent that is licensed by its head (the element that the complement is dependent upon), which means that this particular head element permits the dependent while others don't. Take the sentence

I persuaded my Congressman that he should not support the bill.

Here the (finite) that-clause is licensed by persuaded. If I changed the verb to ask, the license would be denied:

*I asked my Congressman that he not support the bill.

Ask licenses a finite clause with whether:

I asked my Congressman whether he should support the bill.

Ask (as well as persuade) licenses a (nonfinite) infinitive clause:

I asked my Congressman not to support the bill.

Note that not all clauses need licensure. Adverbial clauses may be used freely. Thus I may add the clause

While I was in Washington, DC last month

to a main clause no matter what its verb -- persuade, ask, whatever.

So to consider the example in your question, what are we to make of the that-clause about the average American? Is it licensed by arrogance? The answer from your cite:

Notice that there is no way we can say in general that the noun arrogance takes content-clause complements: it just isn't grammatical to say something like *His arrogance that everything will be all right amazed me. (Try replacing arrogance by assumption and note the difference.)

No license, no complement.

I agree wholeheartedly with the article by Geoff Pullum that you gave a link to. And deadrat is right on the money here too. There's no doubt that although content clauses are often complements, they can’t always be assigned that function, as the examples that Geoff gave in his article clearly demonstrate. His final comment is particularly telling:

"So the property of being a finite complement clause is distinct from the property of being a content clause, despite the fact that nearly all content clauses function as complements".

In any case, clause is a category and complement is a function, so the term 'complement clause' really makes no sense. And worse still, some people even call them noun clauses on the basis of some functional analogy with nouns. The term 'noun clause' is highly misleading. Content clauses aren't nouns; they don’t behave like nouns or NPs; note for example that nouns and NPs don’t normally function as extraposed subject or as complement to a noun or adjective. And consider this example:

“The suggestion that they cheated was quite outrageous”.

The subordinate clause in bold would traditionally be classified as a noun clause, yet it’s not replaceable with a noun. So the analogy fails, and along with the term ‘complement clause', the term 'noun clause' is best avoided.

Incidentally, for the syntactically uninitiated, there are three major subclasses of finite subordinate clause:

Content clause - Relative clause - Comparative clause

The first kind, the content clause lacks the special properties of the other two and is regarded as the default kind of subordinate clause and hence 'content clause' is a very important label indeed.

If someone asked you what category a particular clause belonged to, you wouldn't answer 'complement clause', would you? You'd most like answer something like 'relative' or 'infinitive' or perhaps 'finite'.

Only if you were then asked to describe its role (function) in the sentence would you say 'complement' or 'subject' or whatever. So to call a finite subordinate clause a complement clause is surely conflating category and function.

Surprisingly, we still hear experienced grammarians who should know better using inaccurate terminology like 'complement clause', 'noun clause', 'adjective clause' and so on, whereas those terms belong in schoolboy grammar books and nothing more scholarly.