I think the difference is to best explained in their origins:
unify from unificare from uni + the root of facere 'make' or: one + make = make somethings one
unite from unitus, PP of unire 'unite' from unus 'one' or: 'become one'
From these origins, unify is always the more passive: somethings are made one, unified, whereas unite is more active, we unite - ourselves or others.
'United we stand, divided we fall.' Whoever wrote that did not use unified, because 'we united' we were not 'unified' by some third party.
But the difference is slight, and there is overlap in the usage. 'He unified the country.' 'He united the country' I can't see a difference here.
'We have our family arguments, but, in the face of threat from outside, we unite as one.' I don't think unify would be appropriate here, because 'we' are active, not passive.
So: if you and others actively come together by your own volition, unite. If you and others are brought together, through fair means or foul, by a third party, unify. As a transitive verb, either will do, I imagine.
That's my theory anyway!
