> For them it is about playing pieces perfectly regarding their technique and adding personal expression. Therefore, the choice of fingering is crucial. Jazz pianists, on the other hand, concentrate on the "What". They are always prepared to improvise and adapt their playing to create unexpected harmonies

I minored in music at college and studied both jazz and classical composition/theories. One key takeaway I had from my education is that notes that may considered to be dissonant in classical music are in many cases acceptable in jazz, given the chordal context and how a player chooses to resolve it as progression goes.

I'd even expand a bit further and say that not only musician's brains work differently, but also brain activities of the audiences might differ when listening to jazz and classical pieces. Typically a classical audience might be a bit uncomfortable hearing a dissonant note and treat it as a "wrong note" while a jazz audience might think a dissonant note as being temporarily "out of box" to create tension and expect it to be resolved later.