How does it come about that specific areas of the brain are associated with specific functions? During the development of the human brain, specific areas come to perform specific functions. How (and when) does this differentiation come about?

Presumably, some areas of the brain naturally take on a function that is based on their physical proximity - for example, the visual cortex is the brain area directly connected to the optic pathway. Hence from a developmental perspective, it is natural for it to take on the role of processing visual information.

However, the production of language, for example, could potentially take place in any area of the brain adjacent to the auditory cortex. So how does it come about that it typically always gets processed in Wernicke's area?


Is it predetermined genetically? And if so, how? (pre-mapped synapses?)
Or is it determined later through some kind of learning process?  And if so, what is the explanation for the similarity of brain maps between individuals?
Short answer
Neural wiring is governed by nature and nurture. 

Background
I'm not a language person, but I will try to address the question on more familiar grounds to me, namely sensory systems.

A bunch of most intriguing studies have addressed your question directly. Among these is the study from Frost &amp; Metin (1985), who severed the optic tracts of newborn hamsters and found that optic axons would project to the somatosensory thalamic (ventrobasal) nucleus instead. The experimentally induced retinal projection to the somatosensory nucleus occurred through the stabilization of an early, normally transient projection. Moreover, visual stimulation reliably evoked neural responses in the primary and secondary somatosensory cortices (SI and SII) in treated hamsters. Not only that - the representations of the visual field in SI and SII showed a partially retinotopic organization. In other words, the animals had effectively converted (part of) their somatosensory cortex into visual cortex.

Later, Sur et al. (1988) showed with a similar procedure that retinal cells could be induced to project to the medial geniculate nucleus (MGN), the principal auditory thalamic nucleus, in newborn ferrets. They showed that many MGN cells were then visually driven and that the auditory cortex became visually responsive. Some visual cells in auditory cortex were direction-selective or orientation-sensitive, resembling the complex cells in primary visual cortex. 

Hence, functional visual projections can be routed into nonvisual structures in higher mammals, suggesting that the modality of a sensory thalamic nucleus or cortical area may be specified by its inputs during development. However, there is a short time window where the extent of neural plasticity will allow such gross abnormalities to develop - both studies used newborn animals therefore.

References
- Frost &amp; Metin, Nature (1985); 317(6033): 162-4
- Sur et al. Science (1988); 242(4884): 1437-41