Upvote:1
Hinduism teaches that each person has an immortal soul or self that reincarnates and transmigrates after death. It changes body like a person changing clothes.
As a person sheds worn-out garments and wears new ones, likewise, at the time of death, the soul casts off its worn-out body and enters a new one.
Hindu text BG 2.22
Buddhism teaches that all phenomena is not self (sabbe dhamma anatta) - Dhp 279. Please see this answer for details.
From moment to moment, there is birth, death and (re)birth of chained conditioned processes until final liberation.
The mental idea of the self (that arises from the chained conditioned processes) has renewed existence through transfer of information, as long as craving has not ceased.
The quote below explains this transfer of information using analogies.
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.".
Milindapanha 3.5.5
Upvote:2
One of the most fundamental differences is that Hindus believe in a soul (atman) that passes from one life to a next, like when "you die", "you" will be reborn in heaven. Buddhists know this to be untrue. What dies and what arises is just formation of aggregates which in of themself are products of the flow of karma, not any sort of "self" that is within a soul.
Some later (Mahayanist) Buddhist teachers have tried to mistakenly proport this atman concept, most noticibly asanga, though this is not the Buddhas teachings, Anatta, non self is at the fundamentals of Buddhas teachings and in turn at the fundamental differences between Hindu and Buddhist understandings. Though more often than not the explanations of realms and existences in the cosmology of both sects coincide. So heaven and hell exist in both for eg.