Rebirth and Reincarnation are, in essence, the same thing, but with a key difference: choice.
In Rebirth, a soul's next life is determined by its Karma; whereas in reincarnation, the soul can decide where and to whom it will be born. This happens because the more advanced a soul is, the more specific learning experiences it requires so, in a sense, we all go from rebirth to reincarnation over time.
What we sometimes forget, is that time is not linear for the universe.