What are the factors that contribute to genetically identical individuals behaving differently?
Your question has actually more to do with developmental biology and the origin of phenotypic variance in populations than with neuroscience.
Here is a list of factors which variance explain observed phenotypic variance in a population:

Genetics (incl. additive, dominance and epistatic variances)
Environment (incl. macro and micro environments)
Developmental noise (incl. cellular noise)
Epigenetic
All the covariances among above factors

Of course, when you consider a pair of monozygotic twin as your population, there is little to no genetic variance. They will also likely experience little epigenetic variance. Assuming they live together, practice the same sport, eat the same type of food, etc... then they will also experience little macro-environmental variance. Micro-envrionmental variance and developmental noise are probably the main factors causing these two hypothetical twins to differ.
You might want to read the post Why is a heritability coefficient not an index of how “genetic” something is? to better understand the current post.