In Buddhism, the word rebirth is used quite liberally by the translators. For example, if we examine some of the suttas, we will find various words used for rebirth, such as:
Upapannā = found in MN 4
'These beings — who were endowed with bad conduct of body, speech & mind, who reviled noble ones, held wrong views and undertook actions under the influence of wrong views — with the break-up of the body, after death, have re-appeared in the plane of deprivation, the bad destination, the lower realms, in hell.
Opapātikā = found in MN 117, which actually means 'spontaneous birth'
And what is the right view that has effluents, sides with merit, & results in acquisitions? 'There is what is given, what is offered, what is sacrificed. There are fruits & results of good & bad actions. There is this world & the next world. There is mother & father. There are spontaneously reborn beings; there are priests & contemplatives who, faring rightly & practicing rightly, proclaim this world & the next after having directly known & realized it for themselves.' This is the right view that has effluents, sides with merit, & results in acquisitions.
Upapajjati = reappears Paccājāyati = reborn = found in MN 135
"Here, student, some woman or man is a killer of living beings, murderous, bloody-handed, given to blows and violence, merciless to living beings. Due to having performed and completed such kammas, on the dissolution of the body, after death, he reappears in a state of deprivation, in an unhappy destination, in perdition, in hell. If, on the dissolution of the body, after death, instead of his reappearing in a state of deprivation, in an unhappy destination, in perdition, in hell, he comes to the human state, he is short-lived wherever he is reborn.
Upatti = found in MN 120
Those intentions and that abiding, developed and made much, conduces to reappearing there. This is the path and method to be born there.
Upapajjati = found in MN 57
"Here, Punna, he develops the dog duty fully & unstintingly, he develops the dog-habit fully & unstintingly, he develops the dog mind fully & unstintingly, he develops dog behavior fully & unstintingly. Having done that, on the dissolution of the body, after death, he reappears
in the company of dogs. .....................................................................................
Sopapajjati = found in Dhammapada
137. He who inflicts violence on those who are unarmed, and offends those who are inoffensive, will soon come upon one of these ten states:
138-140 Sharp pain, or disaster, bodily injury, serious illness, or derangement of mind, trouble from the government, or grave charges, loss of relatives, or loss of wealth, or houses destroyed by ravaging fire; upon dissolution of the body that ignorant man is born in hell.
(Kāyassa bhedā duppañño, nirayaṃ sopapajjati)
Thus, if we examine what all these teachings have in common, we will find in Buddhism, rebirth is a consequence of one's actions whereas reincarnation is a meta-physical doctrine, of the soul going from one body to another. For example, the Bhagavad Gita states:
For the soul there is never birth nor death. Nor, having once been, does he ever cease to be. He is unborn, eternal, ever-existing, undying and primeval. He is not slain when the body is slain.
As the embodied soul continuously passes, in this body, from boyhood to youth to old age,
the soul similarly passes into another body at death.
A self-realized soul is not bewildered by such a change.
In short, rebirth is a moral or kammic principle whereas reincarnation is a meta-physical principle.