Since Quora (predictably) deleted this question (no wonder I hardly write there nowadays), I am reposting my answer here:
Why are women, as a whole, more neurotic than men?
Costa, Terracciano and McCrae found that across 26 cultures (with 23,031 people studied), women were more neurotic than men (in the Big Five personality traits).
Neuroticism here is defined as the tendency to experience negative emotions, or emotional instability, and neuroticism is actually associated with psychological disorders (so I'm not sure why you think there's a contradiction between women suffering from more psychological disorders and their being more neurotic than men).
Multiple researchers ((deleted Wikispaces link), The Evolution of Personality Variation in Humans and Other Animals, The Association Between Risk Taking And Personality) conclude that neuroticism is correlated with risk aversion; if you are afraid of risks you're going to spend more time worrying about them, which is/leads to neuroticism.
Meanwhile, it has been found that women are more risk averse than men. Suggestively, it seems that this sex difference also exists in animals (BIOLOGICAL BASIS OF SEX DIFFERENCES IN RISK AVERSION AND COMPETITIVENESS).
This relates to Triver's theory of parental investment (Tool Module: Sexual Selection and the Theory of Parental Investment). Basically women take more risks in reproduction, so they will be more risk-averse than men (because risk loving women did not survive to reproduce).
Addendum: We also know that neurotic women have more children, which supports the above.
IIRC the other answers spouted the usual liberal line about patriarchal oppression making women neurotic.