This is a difficult question to answer, because both aches and pains are subjective experiences - like colours - which you're unable to share, but assume everybody understands. I would have assumed that every language has words for both ache and pain, so a dictionary would tell you the difference in an instant. But, I would also assume you've done that, so your native language might not (I'm interested to find out what language that is).
An ache is a persistent discomfort, typically dull so that you can try to ignore it, but sometimes all-encompassing, yet not sharp enough to describe as pain. Your legs would ache after a tough run; you would not describe this as pain. You usually get a headache, not a head pain. You would suffer pain when you cut your finger, then experience an ache as the wound heals.
A pain is something more localised, often (but not always) short-lived, and something you'd be less able to ignore.
When you receive an injection, there is a pain as the needle goes in. During the following days, the surrounding area will ache.
There is considerable overlap between the two, and it would be quite acceptable to say "the ache in my shoulder is painful".
Poets and songwriters quite often speak of their heart aching. This fits well with a persistent sense of yearning or melancholy. If they said there was a pain in their heart, it would suggest a quite different emotion.
Describing pain and discomfort is difficult and subjective; I imagine in any language. When a doctor asks you how much something hurts, how can you explain in a reliable way?