They are similar. The way I see it is: "To vanish" - means something that happened quickly and unexpectedly. "To disappear" - Something that is gone or missing however not like vanished where it was unexpected.

These two words are very close synonyms, and are almost always interchangeable. I agree with Richard that 'vanish' is more sudden, but not about whether the event is unexpected or not. Something can certainly disappear unexpectedly - in fact most disappearances are unexpected. I think the only difference is that 'vanish' sounds more dramatic.