I'm no expert but from what I understand, Montessori is about using things the "correct" way while Reggio is all about free exploration of materials.
I love Reggio but my personality is SO Montessori! I keep biting my tongue so I don't say "That's not how to use that!" "Those toys don't belong there!" "That's for __________, not ______________"
I'm too much of a rule-follower, LOL!