I have been asked this question many times. Since the Moon’s qualities are that of maternal care, affection and feelings, the question is appropriate.
We must consider first how strong the Moon is in a woman's natal chart. A Moon with few aspects is not going to be much important in the person's traits in any case. But if the Moon has strong aspects or is in a congenial sign such as Cancer or Pisces, then it will inspire the personality in a factual way.
Then we need to consider that the Moon is quite reserved, she doesn't show off her nature loudly. Is difficult to interpret the Moon from the external appearance or from a superficial encounter.
Since the Moon is more connected to the feelings and the inner world instead of the ego and the will as the Sun is, it takes a certain degree of intimacy to become aware of the Moon's qualities in a person.
Furthermore, we should be aware that the Moon doesn't display her qualities much in “actions,” but in “reactions.” Being more sensitive and passive than the Sun, it manifests mainly through emotional feedback, which might not even show on the surface, especially in case the Moon stays in more introverted signs, in discreet houses or has aspects with withdrawn planets, as for instance with Saturn.
The Moon reveals what happens inside, while the Sun shows what we do on the outside. If we agree that women have a broader inner world than men, then for sure the Moon is going to be more significant for women.
Through the Moon and Sun dialectics we can understand better what's the outside personality and the inside feelings or reactions. A vulnerable Sun in Cancer, on a better outlook, might unfold a strong Moon in Aries. Or an easygoing Geminis with the Sun in the 11th house might have a hidden deep Moon in Scorpio in the 8th house, which would be revealed to others only when the person feels substantially safe, understood and comfortable.
Especially in the past, some astrologers said that when the Moon is in signs like Aries or Capricorn, it carries away femininity, but we need to agree on what this means. Femininity was and still is much a cultural concept. People have different ideas of it, let's say in Germany or in Brazil, in Morocco or in Japan. I'd say instead that the Moon shapes different ways of being feminine.