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Dark Link was created by Ganondorf for the sole purpose of serving him. He was made so he could kill Link. He waits for you in the Water Temple, but that's it.

Shadow Link on the other hand, is his own person. He has control over a whole Deku Scrub army (not that they like it) and though he also tries to kill you, that's of his own will. Also, there are several of him. He seems to have the ability to split himself to match up with Link and the other 3 parts of him, more even. He follows you around Hyrule attempting to get rid of you by setting fire to you, hammering you, running you through with a sword, or throwing giant timed bombs that cause instant death if you don't find shelter.

Also, Dark Link has no mind of his own. If you were to read the manga, he charges blindly at Link in an attempt to be rid of him.

Shadow Link in the FS manga controls armies of Darknuts and can posses other people(in an off sort of way). Also, he can make himself look like Link, by which I mean changing eye and hair color.

Then there is also the fact that they look TOTALLY DIFFERENT and exist in different times. Dark Link in Ocarina, and Shadow Link in Four Swords Adventures.