What is the conceptual distinction between coordinating conjunctions and subordinating conjunctions?

The FANBOYS mnenomic has been popular in the teaching of English to native speakers (particularly in the USA using textbooks such as Warriner's English Grammar and Composition). One of the goals of instruction is to help students learn to write correctly constructed and punctuated sentences, and to avoid sentence fragments and run-ons or comma splices.

In such a context it is useful for students to know that they can construct a complete sentence containing one clause beginning with a FANBOY (coordinating) conjunction, but not beginning with a subordinating conjunction:

• But his plan failed. [OK]
• Because his plan failed. [Wrong - fragment]

Of course, this is a simplistic approach. And descriptive grammars such as The Cambridge Grammar of the English Language have shown that the FANBOYS are far from a homogenous grouping. But many teachers feel that they are useful mnemonic for younger learners.

As to your question, you ask:

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?

You are certainly right to point out the explanatory circularity and that removing both coordinating and subordinating conjunctions results in correct, free-standing clauses. His plan failed in the case above. Nevertheless, there is one essential difference between the two types.

Clauses beginning with a subordinating conjunction (i.e. dependent clauses) can be placed before or after the independent clause:

• His plan failed because he had overlooked an important detail.
• Because he had overlooked an important detail, his plan failed.

On the other hand, clauses beginning with a coordinating conjunction (i.e. independent clauses) must follow the previous independent clause:

He had overlooked an important detail, so his plan failed.

*So his plan failed, he had overlooked an important detail is ungrammatical.

Extrapolating to your examples, you can say both:

• Stock prices would have plummeted, unless the company had issued a public apology.
• Unless the company had issued a public apology, stock prices would have plummeted.

But only:

Stock prices would have plummeted, but the company issued a public apology.

and not:

*But the company issued a public apology, stock prices would have plummeted.

I have also wondered what exactly makes one subordinating vs. coordinating.
BUT keep in mind that 'ing' at the end of "subordinating". The conjunction itself subordinates or coordinates a clause. It is not something about the nature of each clause, but what the conjunction does to the clause.

"Stock prices plummeted and the company issued an apology."
"Stock prices plummeted although the company issued an apology."
Even though this particular "and" would sound odd if you reversed the order, because there is an implied time sequence, still, if one clause is more essential than the other, it is not because the conjunction tells us so. In the case of 'because', 'although', 'if', 'when', 'where', and I'd say 'unless', on the other hand, that one word tells us that one clause is the point and the other is a detail or a condition. That's NOT because one of the facts mentioned is inherently less important, but because the conjunction makes it less important.

"I slept well because I had just worked for 16 hours trying to separate a pair of conjoined twins."

Obviously, the main clause "I slept well" is far less interesting, but that conjunction alone makes the other part "subordinate".