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free jazz is more of a in the rejection of notes and rhythms in a jazz idiom and (at least since the 70's) has a tradition of sorts based on the seminal recordings of trane, ornette, cecil, etc. (even though this tradition is a sort of oxymoron since the original free jazz rejected tradition).
avant jazz is music that's artier and a more educated kind of improv that are based on textures and have a purpose or might even include notes and other elements of established music like notes or rhythms, done with jazz instrumention primarly. its origins are more in the 20th century classical music but it also has origins in free jazz and have many other styles of music mixed in (like ambient, drone, etc.).
examples:
peter brötzmann - machine gun = free jazz
otomo yoshihide's new jazz orchestra - out to lunch = avant jazz
avant jazz: check out nihilist spasm band!!!
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