Raphael is the most aggressive and constantly fighting with Leonardo, so he gets a weapon that's intentionally defensive and made to counter swords.
Leonardo is the responsible leader, so he gets the most obviously deadly weapon.
Donatello is the smartest, and often non violent, so he gets the most simple weapon so he doesn't over think things. Yet at the same time, a staff is surprisingly versatile.
Michelangelo has ADHD but a lot of natural ability, so he gets a weapon that requires a lot of focus and coordination to use effectively to tamper his short attention span, while also making use of his need to showboat.