Paramatman, the supreme Self, names the highest reality considered as self, as distinct from jivatman, the individual self embodied in a particular life. The distinction is given its classic image in the Upanishads: two birds, companions, perched on the same tree, one eating the sweet fruit while the other looks on without eating (Mundaka 3.1.1, repeated at Shvetashvatara 4.6). The commentarial traditions read the eater as the individual self entangled in experience and the watcher as the supreme Self, present as witness.
What the relation between the two actually is constitutes the central disagreement of Vedanta, and the schools are defined by their answers.
Advaita, in Shankara, holds non-duality: the individual self is not a part, a portion or an image of the supreme Self but is identical with it, and the appearance of difference is the work of ignorance. Vishishtadvaita, in Ramanuja, holds qualified non-duality: selves and matter are entirely real and constitute the body of Brahman, so that the individual self is genuinely distinct while wholly dependent, related to the supreme as an attribute to its substance or a body to its soul. Dvaita, in Madhva, holds outright difference: the supreme Self and individual selves are eternally distinct, and complete dependence does not entail identity, Madhva argues that dependence in fact presupposes distinction. Bhedabheda schools hold difference and non-difference together, in various ways.
The dispute is not verbal, because different practices follow from each answer. If the self is already identical with the supreme, then liberation is the removal of an error and knowledge is the means; discrimination and enquiry are the discipline, and devotion is preparatory. If the self is genuinely distinct and dependent, then liberation is a relation of loving service that grace establishes, and devotion is the means rather than a preliminary to it. This is why the theistic schools developed bhakti as primary and Advaita developed viveka.
All the schools read the Bhagavad Gita, and each finds its position in it, which is a large part of why the commentarial literature on that text is so vast. Samkhya, standing outside Vedanta, uses the vocabulary differently again: in its classical form it is non-theistic and speaks of a plurality of purushas rather than one supreme Self, and Yoga adds Ishvara as a special purusha untouched by affliction, a further position, and one that shows how contested the ground is even over what the question is.