"Say" or "Tell"

How would you explain when to use each of these words?

In Georgian, there are two words for these verbs -- one that is equivalent to "say" and another to "tell." But, in Georgian, they are interchangeable. You could use either word without changing the word order or sentence structure, and the sentence would still be correct. Not so in English.

Tea is working so hard at using these two verbs correctly, but the difference between the two is not concrete (as least, as far as I can tell). I have looked online for some rules of when to use which one, but my search has not given me anything useful. The best I can do right now is give her examples of proper use:
"Tell me something....."
"Say something...."
"Say what you mean."
"Tell me what you mean."
"Tell her the story."
"What did you say?"
"What did she tell you?"
"What did she say to you?"
"She said that she would be here."
"She told me that she would be here."
"Tell me the answer."
The best "rules" I can come up with so far is that usually a pronoun is used after "tell," but not after "say." That's not very specific. And the words are not merely interchangeable; however, they can often be used to express the same type of thought -- like "Say what you mean," and "Tell me what you mean." The sentence doesn't change meaning with the change in the verb.

Tea and I were discussing these two verbs again tonight as we baked a giant chocolate cake for a suphra we are going to tomorrow. We were trying to come up with general guidelines for when to use which word. As we talked, Tea got pensive for a moment. Then she remarked that it is amazing how thought-provoking language is when you start swimming in the entirety of it.

This is so, so true! The complexity of language is staggering. It is so much more than vocabulary and grammar. Reading rules, spelling, synonyms, homonyms, antonyms, idioms, homophones, interrogatives, sentence structure, punctuation, capitalization, syntax, semantics, rhythm, intonation, pronunciation, figurative language, and the list could go on and on.

Learning the nuances of any language takes years and years of speaking it, listening to it, thinking in it, being surrounded by it -- as Tea said, "swimming in it." I know that sometimes she feels like she is drowning in it!

Good thing I used to be a lifeguard.