The typical way to highlight the difference in translating into English, is to say Buddhism teaches rebirth of causes & conditions not identity, whereas Hinduism teaches reincarnation, a new instantiation of a persistent identity. As I understand it though, they use the same Sanskrit words meaning rebecoming or reborn (punarbhava or punarjanman).
I wish more people, including many Buddhists, would ask this question. Anatta as one of the three marks of existence, is at the core of Buddhist analysis. Whereas, the elaborate cosmology of types of being and realms is never expounded in one place, but drawn together long after the Buddhas time from fragmentary references throughout the sutras.
Whenever people asked the Buddha questions about the soul he would ask exactly what is meant, then answer in those terms. This is a good practice, because the simple word easily misleads based on people's different preconceptions. Buddha taught a middle way between annihilationism & eternalism, and taught that either view is seriously problematic. The first corresponds to simple materialism, that our life ends completely with the body. The latter to metempsychosis of an eternal soul.
Rebirth is explicitly pictured as a bad thing in Buddhism, the opposite of the aim of practicioners. Good actions and ways of thinking can lead to more positive rebirth, but this is always pictured as a poor substitute & distraction to awakening to the true nature of things, severing the chain of dependent origination which begins with ignorance (of the true nature of things).
The metaphor for awakened I like to use is drawn from understanding why we are polite. We could see it as motivated by reward: expecting politeness & consideration from others. But among strangers we are very unlikely to see again, we continue to be polite. It is part of manifesting the kind of social world we want to see. And we shift motivation from future reward or condemnation, to the acts themselves, as embodying living the kind of person we want to be. The Eightfold Path is a process of cataloguing and addressing how we live with this same approach, to go from acting for specific results, which tie an action to a time frame, even if one outside of our own lifetimes before it will reach fruition. In the politeness example, it might mean going beyond acting to create the social world we want to be in, to a deeper awareness of how our actions shape our mind, who we are right this moment only (meditation being a key route to deepening that), and so go beyond trying to create the society we can imagine we want, to drawing on deeper guidance, towards manifesting what we cannot imagine, but aligns with who we are becoming.
A well known Buddhist metaphor for rebirth without transmigration:
The king asked: "Venerable Nagasena, is it so that one does not
transmigrate and one is reborn?"
"Yes, your majesty, one does not transmigrate and one is reborn."
"How, venerable Nagasena, is it that one does not transmigrate and one
is reborn? Give me an analogy."
"Just as, your majesty, if someone kindled one lamp from another, is
it indeed so, your majesty, that the lamp would transmigrate from the
other lamp?"
"Certainly not, venerable sir."
"Indeed just so, your majesty, one does not transmigrate and one is
reborn."
"Give me another analogy."
"Do you remember, your majesty, when you were a boy learning some
verse from a teacher?"
"Yes, venerable sir."
"Your majesty, did this verse transmigrate from the teacher?"
"Certainly not, venerable sir."
"Indeed just so, your majesty, one does not transmigrate and one is
reborn."
"You are clever, venerable Nagasena."
-from The Questions Of King Melinda, Miln III.5.5. Canonical in Burmese Buddhism but not other traditions, and widely translated and read in the Buddhist world
It should be noted that 'the precise playing out of karma' is given in the Anguttara Nikaya as one of The Four Imponderables (acinteyya), issues that should not be thought about, since this distracts from practice, and hinders the attainment of liberation. This is explicitly not the point of the teaching.
In the Kalama Sutra, explicitly a teaching to those who are truth seekers but not yet Buddhists, Buddha expounded the Four Solaces to be found from the teaching of rebirth by one "who has such a hate-free mind, such a malice-free mind, such an undefiled mind, and such a purified mind" - explicitly focusing how to begin the process of practice aimed at there, a mind like that, and detailing how to make the teaching of rebirth a help, not a hindrance. Because:
"Mind precedes all mental states. Mind is their chief; they are all
mind-wrought. If with an impure mind a person speaks or acts suffering
follows him like the wheel that follows the foot of the ox.
Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with a pure mind a person speaks or acts happiness
follows him like his never-departing shadow.
-opening lines of The Dhammapada
Perhaps the dangers of a 'pure materialist' view can be seen by suicide, which might be a logical choice if life is accepted as having suffering, anguish, unsatisfactoriness as it's in some way 'core qualities'. Such a death by suicide is not the end of it's causes and conditions. Here drawing into the 'present moment only' of such a choice could be misleading. Suffering is passed on to others, and ripples outwards. But also we make a template by how we choose to respond, for how others respond. By not seeking to reconcile that situation, it is allowed to pass on, to recur, and the opportunity to find learning that could go beyond it, is passed. There are transferable tactics or ideas that can help, and be spread like the lamp's light in the dark, in the passage above. It is interesting to note the cross-cultural & cross-era emphasis we place on people's mental state at death, whether promises kept, feuds ended, bucket-lists ticked. Something meaningful can be attained even in last moments that can change the whole tone of a life, and what others draw from it, into their own.
The Dalai Lama is considered an incarnation of the bodhisattva Avalokitesvara, and so have been some other individuals, alive at the same time as holders of that lineage - but is also given the epithet the bodhisattva of 'a thousand hands and eyes', which is to say those of all compassionate people. It seems valid to question whether the Buddhist 'causes and conditions' that are reborn might be expected to be in the form of one future person, in linear descent in time - ie, is this just a metaphor?
Here I would draw attention to the Buddhist approach that mind is in a sense fundamental (this is most associated with Yogacara, the philosophy of Mahayana schools like Zen & Tibetan styles, but note the Dhamapada line: mind precedes all mental states, mind is their chief; they are all mind-wrought). I see this as an assertion about subjectivity, and narration: our experiences are from a point of view, and their impacts go out like that too, into other subjectivities.
Buddhism deconstructs the conventional self, the false intuitions we have that there is something unchanging and unique about us. And I would say leaves this self, the intersubjectivity of our experiences, the way the choices we face are in some measure what everyone faces, and our lessons in some way, lessons for all of us. The ancient metaphor Indra's Net is the best picture of this I know. There is not a true objectivity to the world because no one can experience that, there is instead this compound of subjectivities, reflecting each other: intersubjectivity. In the suicide example, it's not the future consequences, but the manifesting a way to be which will be a template for others, which is the problem, in the Buddhist view, I would argue - accepting suffering for oneself, accepts it on the behalf of others, manifesting resolving it reflects that out instead, not as an abstract, but as an invitation to experience (I was talking recently about how poetry is often more useful in facing death, than is philosophy - inviting us to recite, participate, enact)
I would identify intersubjectivity as key to understanding Kant's categorical imperative, and Rawl's theory of justice, and to an understanding of how communication happens in Wittgenstein's perspective of no private language, and to mirror neurons in our cognitive theory of learning.
Apologies for a long post. As a practicioner of Zen this has been an important topic for me which I have reflected on for a long time. I hope my little rumination is helpful.
Namaste