My idea of the Neanderthal psyche is informed by some paleoanthropological assertions.
The Neanderthals lived in smaller groups than Homo Sapiens and research by anthropologist Robin Dunbar (famous for the 'Dunbar Number', a figure representing the maximum number of relationships an individual can meaningfully sustain) corroborates that Neanderthals were not capable of sustaining as large social groups or as many social connections as modern Humans. Their ability to produce speech and language is hotly debated but probably did not reach the level of sophistication of Homo Sapiens, especially given that their smaller and more cohesive groups, similar to those of the modern apes, did not require sophisticated language. Smaller frontal lobes affected their ability to 'mentalize' or to form interrelated beliefs about what is going on in the minds of others. The Neanderthals would not be capable of following a plot like that of Shakespeare's Othello where the audience must believe that Iago intends that Othello imagines that Desdemona is in love with Cassio and therefore persuades Othello that Cassio reciprocates Desdemona's love.
In keeping with limitations in ability to mentalize and smaller groupings, Neanderthals probably lacked 'cheater detection'. Lying is less profitable in smaller social groups where a liar/cheater is more easily discovered and where being ostracized would be a death sentence, especially in the challenging environment the Neanderthals lived in. As a result, they were probably honest social literalists who assumed honesty in others and could not expect ulterior motives or deception.
The Neanderthals were also not as innovative as their Homo Sapiens counterparts. Whereas the Homo Sapiens toolkit changed dramatically over time, the Neanderthals continued to produce the same tools in the same manner for thousands of generations. It is speculated that they were great apprentice learners, capable of mimicking what they were taught, but not dynamic and open-minded innovators, which requires abstract/symbolic thinking to identify problems and possible new solutions as well as the hypothetical results and second-order effects of these solutions. Perhaps just as Neanderthals would have had difficulty tracking the plot of Othello, they would have been confused by symbolic thinking, though maybe not unlike how most Humans become confused with the abstractions of algebra. To us, the Neanderthals might even appear neophobic and dogmatic.
While their inability to detect deception and be innovative might cause the Neanderthals to appear to be lacking in all symbolic or abstract thinking, they did bury their dead along with flowers and grave goods, indicating a spiritual component to their understanding of the world. It is conceivable that ideas of an afterlife might have been spurred by dreams in which dead loved ones would appear and interact with the dreamer. Given some of their psychological limitations in understanding abstraction, it is possible that the Neanderthals did not have a clear mental demarcation between the literal waking world and the dream world, although this is purely my own speculation.
As for their Homo Sapiens counterparts, it is impossible to know what the Neanderthals thought of their cousins. Stephen Jay Gould, J.B.S. Haldane, Desmond Morris and others have theorized on the role neoteny, the retention of juvenile traits into adulthood, has had in the evolution of Homo Sapiens. Longer childhoods (and juvenile traits in adulthood) postpone sexual productivity but confer advantages in learning ability (as juvenile mammals are more receptive learners) and smaller cranial sizes during childbirth, making childbirth less dangerous to the mother. Unlike their cousins, according to research in teeth development by Fernando V. Ramirez Rozzi, Neanderthals had shorter and much faster childhoods, reaching adulthood by age 15. A shorter juvenile development period, more akin to the apes, may have limited their acquisition of learned knowledge and necessitated larger craniums along with the attendant risks in childbirth.
The features of Homo Sapiens, including less pronounced brow ridges, smaller noses, flatter faces, and generally softer, neotenous features might have made modern Humans appear child-like to the Neanderthals.
Edit:
Perhaps just have someone narrating the story/feelings/expression?
I am cautious about being too heavy-handed with it, though. Maybe I will do this.[This message has been edited by Al_Kharn the Great (edited 06-20-2014 @ 07:39 PM).]