These phrases/words are in most cases interchangeable, but here are some examples that might help.
He remembers to buy apples. (He remembers that he must buy apples and then buys them.)
✓ Han kommer ihåg att köpa äpplen.
✖ Han minns att köpa äpplen.
He remembers buying apples. (He remembers the act of buying apples which he already performed at some previous time.)
✓ Han kommer ihåg att han köpte äpplen.
✓ Han minns att han köpt äpplen.
Rule of thumb: "Minnas" is use for the past / "Komma ihåg" is used for the present & future.
But since the Swedish language is so weird and filled with exceptions you can never be sure :P
Unless you're in a *very* formal situation, misusing these won't matter that much.