It is because of the composition of the jury pool. The jurors in Westchester are mostly property owners who are very conservative whereas the jurors a few miles a way in the Bronx are mostly members of a minority group, more likely to be renters, and far more liberal in awarding damages. A mediation requires both sides to agree. If you do not like the offer you can always proceed to trial.
The issue is what would have been different had you been told right away. Medical malpractices are very expensive to bring so most lawyers won't take a malpractice case unless there a significant injury. Having to take medicine probably won't be sufficient for a lawyer.