Nothing. They are often pronounced the same, although in some parts of the US "route" is pronounced to rhyme with "out".

"Root" has the more traditional "oo" sound, as in "boot," or "suit." "Route" rhymes with "about" and has the same vowel sound as in "ouch".

I think it also depends on the form of the word. Around here (California), we say route like root most often (though I've heard it and used it the other way), but when an -ing inflection is added to the end, it always ends up rhyming with out. For example, the "routing" in "routing number" rhymes with "outing." An operator might tell me they're routing (rhymes with outing) my call (if it's 1980, anyway :p).