While it is easy enough to identify coordinating and subordinating conjunctions by memorizing lists of them, the actual distinction seems arbitrary. Coordinating conjunctions are said to join (among other sentence elements) two independent clauses, while subordinating conjunctions are said to join a subordinate clause to an independent clause.
The problem is that it appears to be the very presence of a subordinating conjunction which makes an otherwise-independent clause subordinate. For example, in the sentence...
Stock prices would have plummeted, unless the company issued a public apology.
...the second clause is only subordinate by virtue of including the word "unless." "The company issued a public apology" expresses a complete thought, but "unless the company issued a public apology" does not. So the presence of the word "unless" explains why the second clause is subordinate, which explains why "unless" is a subordinating conjunction.
However, in the sentence...
Stock prices would have plummeted, but the company issued a public apology.
...the same analysis is not applied. Rather, the word "but" is considered to stand between the two clauses, belonging to neither. This grants independent status to the second clause, "the company issued a public apology," whereas if the word "but" were to be included, it would be a subordinate clause, and therefore "but" would be a subordinating conjunction.
A sort of explanatory circularity is apparent, and it's not at all clear to me why "unless" and "but" should be treated differently in these examples. What is so special about the FANBOYS conjunctions that they are said to stand outside of independent clauses, while the rest are said to be part of otherwise-independent clauses, making them subordinate?