A miner is someone who works in a mine. Minor means lesser or less important, and as a noun means a person under the legal age of adulthood. One is an occupation, the other a comparison — and in law, a category of person.
The core difference
An occupation and an adjective that doubles as a legal noun.
- miner — someone who works in a mine: coal miners.
- minor — lesser: a minor injury; under legal age: serving alcohol to a minor; a musical key.
The musical sense
Minor keys and chords are the ones conventionally described as sad or darker in character, as against major. That is a third distinct use, and it is why "a minor chord" and "a miner's chord" are amusingly different things.
The academic sense
In American education, a minor is a secondary field of study alongside a major. Another use of the lesser sense, and one that has no British equivalent.