What is the difference between a "strong" (stable) nervous system from a "weak" (unstable) nervous system from the point of view of neurology?

Nothing, from the point of view of neurology, there is no such gradation. In practice, there is a certain individual level of resistance to external and internal stimuli, at the intersection of which various stress reactions are possible, however, except on the basis of experience, this level cannot be determined, and in one person it will fluctuate throughout life.

Mental stability is always a combination of genetics and external factors, it is impossible to unequivocally identify a weak nervous system as a disease (yet?). But there are regulatory systems responsible for the response to stress: catabolic hormones (cortisol), which are produced during intense stress and spur the body, acting for a short time, and anabolic (testosterone, growth hormone), which restore the body to normal, preparing for the next stress ... Normally, the ratio of anabolic to catabolic in the body is high, which allows a person to quickly recover physically and mentally. Accordingly, in people with a weak psyche, this ratio is lower.

Also, with low stress, amines are released into the bloodstream, which act as cortisol, partially replacing it, this is less stress for the body than the release of cortisol. Emotionally stable people are more likely to produce amines (rather than cortisol) and their levels are higher in a calm state than in non-stable people.