So what is LTT then? He does have the knowledge and skills of LTT, Rand can now draw because of him and he knows the faces of the Forsaken because of him, not to mention all the different weavings for channeling. Is it some dormant part of Rand's soul remembering the old persona?
Happyman's explanation is right, of course, and one closely mirroring the explanations given by Robert Jordan himself. To add to what Happyman wrote, imagine a soul as a multifaceted diamond, where the facets represent the various personalities associated with the soul over the millennia. Such a facet would then contain every aspect of an incarnation of the soul in question, including personality traits, memory and other idiosyncracies. Normally, these facets are clearly demarcated and separate from each other. In the case of Rand, the boundary between his personality facet and Lews Therin's has been deteriorating, until the two actually come into contact with each other. As indicated by Semirhage, and all extant evidence suggests that she was not lying, this was a known form of schizophrenia in the AoL. That is, it is a variant of the multiple-personality disorder, only with the distinction that the imposing or intruding personality is one which had been previously associated with the soul of the afflicted person. That is, the voice, memories and characteristics of Lews Therin are real, but arise as a consequence of a known mental disease. We have every reason to believe that this breaking down of personality boundaries has come about as a direct consequence of the taint. We know that taint madness takes a variety of different forms, that is, there is no fixed taint madness by which all male channelers are beset. As I see it, the taint merely accentuates or brings to the forefront mental diseases to which a person may be predisposed.