SOCRATES VERSUS BUDDHA ON THE SOUL
If Buddhism denies the existence of any continuing self or soul, this appears to conflict with Socrates' view of a continuing soul which is freed and released from the regions of the earth as from a prison. The soul continues to exist, Socrates says, but in radically different conditions. For Buddhism there is no soul to continue to exist.
SOCRATES VERSUS BUDDHA ON DEATH
Socrates sees death as the precondition for the soul's new freedom. In Buddhism, however, the correspondingly blessed state can occur before death. The Buddha entered nirvana in his 35th year, long before his death.
SOCRATES AND METEMPSYCHOSIS
Metempsychosis is popularly defined as the
belief that at death the soul passes into another
body. But one cannot be long engaged in studying the doctrine as it appears in Greece and India
before discovering that the popular definition can
be rendered more precise by the addition of three
restrictions: the place where the soul and its new
body dwell must be, at least in part, this world;
the new body must be acquired for more than a
temporary [? brief] period; and the soul, which passes
from one body to another, must be that which
creates an individual [? must be the same soul over time]. (Herbert S. Long, 'Plato's Doctrine of Metempsychosis and Its Source', The Classical Weekly, Vol. 41, No. 10 (Feb. 16, 1948), 149-155: 149.
Plato seems to have derived the idea of metempsychosis from Pythagoras but this leaves open how Pythagoras came by the idea. (Long, 151.) Metempsychosis as such does not imply immortality; extinction could intervene at any point. Plato/ Socrates has separate 'proofs' of immortality.
REINCARNATION OR REBIRTH
I take this to be the idea that the soul or self survives death through rebirth in another body. This also does not imply immortality; again extinction could intervene at any point. There appears to be no essential difference between metempsychosis and reincarnation but the terms tend to be associated in different traditions of thought with extra ideas. Long's account of metempsychosis, for instances, includes beliefs which are not present in the bare idea of reincarnation or rebirth.
Herbert S. Long, 'Plato's Doctrine of Metempsychosis and Its Source', The Classical Weekly, Vol. 41, No. 10 (Feb. 16, 1948), 149-155.
Steven Collins, 'Nirvāṇa, Time, and Narrative', History of Religions, Vol. 31, No. 3 (Feb., 1992), pp. 215-246.
Dorothea Frede, 'The Final Proof of the Immortality of the Soul in Plato's "Phaedo" 102a-107a', Phronesis, Vol. 23, No. 1 (1978), pp. 27-41.