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"Different" is used to indicate that two or more things are not the same. "Things" in this case can include almost anything -- people, dogs, houses, pizzas, and French films to opinions, thoughts and presidential candidates.
"Indifferent" is used to indicate that you (the speaker) have no opinion about all these different things. It has a mildly negative emphasis. If you are indifferent, you just don't care, usually because the thing or things being discussed are too mediocre to care about. The less formal way of expressing that something is indifferent would be to say "I don't care" or "I don't give a damn."