There's effectively no difference. They appear in a few games: Zelda 2, Link to the Past (only in the GBA port so i dont think it counts,) Ocarina of Time, Oracle of Ages, Twilight Princess, Spirit Tracks and Link between Worlds. Oracle of ages he is a distraction used by the final boss, Spirit Tracks he is only in a minigame, Link to the past he wasn't in the original so it isn't canon, and Link between worlds he represents the enemy in a multiplayer fight, which is again not significant to the story. So that really only leaves Zelda 2, Ocarina of Time, and Twilight Princess.
In Zelda 2, when he shows up as the final boss, you are intended to "fight yourself to prove your worth" so to speak. It was likely left as a final test by the King of Hyrule who hid the triforce of courage. Although, he is referred to as Shadow Link, which may or may not be different from Dark Link (I don't think there is.)
He appears in Ocarina of time, which I believe is his most known role. In the Water Temple, he appears in an illusionary room when you are believed to be trapped inside. He is a pain to fight, and is actually one of the tougher enemies overall to defeat. He doesn't actually have a role past this which is guarding the Longshot. He's really just a miniboss, but the fight is very memorable because the Water Temple is a huge pain, the pace is slowed down from the confusion of being trapped in a room, along with him surprisingly appearing to fight you.
In Twilight Princess, he is just a visual tool to represent the Interlopers, who were the people responsible for originally attempting to gain control over the sacred realm (precursor to Ganondorf.) There were 3 of them in the flashback and it may hold some meaning, but not necessarily.
So in conclusion, Dark/Shadow Link at best is representing the evil side of Link, and is literally put in as a test for Link, who is fighting himself. He is best kept as an alt costume. In the future, he may become something more than just a mirror of Link, then I would be all for his inclusion, but until that happens, an alt. skin/costume is perfect for him.