Mdtryhht

In the sentence "It's raining, but I'm happy," "but" is a coordinating conjunction. Both of the clauses are independent, right?

However, doesn't "I'm happy even though it's raining" mean the same thing? And yet, in this case, "even though it's raining" is a dependent clause, correct, because "even though" is a subordinating conjunction?

So I'm confused. What's the difference between dependent and independent clauses? I'm a secondary English teacher and I'm not sure how to explain this apparent double standard.

Whether a clause is dependent or independent doesn't have to do with its meaning.

That's where the confusion about but with even though came from. They do mean sort of the same thing.

But that fact doesn't have anything to do with whether they introduce dependent or independent clauses.
Dependent means 'hanging from' in Latin, and the idea is that one clause is the main one,

while others hang from it, and are marked as 'subordinate' (another UP/DOWN metaphor) to it.

This is why syntax uses upside-down "tree" diagrams to illustrate sentence structure.

Just a bunch of phrases and clauses hanging around together, like aerialists on a break.

For instance, the subordinate clause can be like an adverb modifying a verb phrase,

• 
  Because she had already started, she finished it.

or like a direct object of a verb

• 
  I told him that she would finish it.

There are coordinating conjunctions (and, but, or, and (rarely) for) that hook together two independent sentences, and they can go on forever:

• Bill saw Sue, and Mary told Joan, and Joan was mad, but Mary just laughed, or maybe not...

However, except for those, every other clause introducer -- every conjunction or adverb or phrase or marker or relative pronoun or preposition or complementizer -- and there are literally thousands of these in English -- introduces a subordinate, i.e, dependent, clause. They are overwhelmingly more common than independent clauses.

So the vast majority of sentences in English are the kind that are called "complex", because they contain

at least one subordinate, i.e. dependent, clause in addition to a main, i.e. independent, clause.

You can say independent clauses are the ones that have a meaning even though they are said alone. If I say “I am happy” you get that I am happy.
Dependent clauses instead have no meaning if they are said alone. If I say “even though is raining” you get that it’s raining but the sentence has no meaning since there isn’t a main clause.

In the simplest terms.

Two independent clauses:

He eats ham but I eat jam.

Two independent clauses joined by BUT: He eats ham. I eat jam. That is, each can stand alone if you remove the conjunction.

Compare that to:

Two clauses, one independent and one dependent.

He eats ham even though I eat jam.

even though is a dependent clause introducer (see John Lawler above).

Even though introduces the idea that "I eat jam". It is not a conjunction.

[May I say that even though I eat jam, I am not happy? joke]