psychological sequelae might be a word you're looking for if you forgive that it's somehow still neurobiological; it is however, not genetic or developmental or something somebody was born with:
Chronic kidney disease, for example, is sometimes a sequela of diabetes,
and neck pain is a common sequela of whiplash or other trauma to the cervical
vertebrae. Post-traumatic stress disorder may be a **psychological sequela** of rape.
Of course, this is a specific kind of pathological condition (a clinical one that requires attention!), not a general term for "psychological effect".
As Artem pointed out, you will have a hard time separating neurobiology from psychological effects if you want to maintain a reasonable/empirical/scientific approach.
Epigenetic is the word for "an addition to" genetics, so you can find out a lot with that search term. Particularly, behavioral epigenetics might be what you're interested in (note that it's still a molecular basis, just not genetic):
Environmental influences, as opposed to intrinsic influences (but still somehow neurobiological, as the biological system has to detect, interpret and respond to external stimuli for them to be significant) have many different names and are a large part of psychology in general. Richard Wilkinson has an excellent talk on how income disparity leads to unhappiness through the psychological conditions it imposes on the poor (another example of Sequelae):