Is there a difference between the neshamah of a man and a woman?

Is there a difference between the neshamah of a man and a woman?

When it comes to the actual makeup of the neshamah there's no difference, whatsoever. But when it comes to a person's role in perfecting that neshamah, Hakadosh Baruch Hu has given each individual a different way and opportunity to go about doing so. For example, when a woman gets married and dedicates her life to her family, to someone other than herself, she is transforming her very nature. If a mother would retain her selfishness, it would be impossible for her to function successfully in her role. She has her children to worry about; sometimes they may not be feeling well, and it is she who will be responsible for tending to them. She has to prepare food for her family, and they rely upon her for many other things as well. She has to carry a child within her and eventually nurse that child. In short, she has dedicated her life to others. But in order for a mother to be able to perfect her neshamah, she mustn't merely do "acts of kindness" like the gentiles do. Rather, her chessed should be with the intention of serving Hashem! This brings Hakadosh Baruch Hu a great amount of satisfaction, and He therefore exclaims, "By assisting others, you are helping My people and serving Me in a form that is most desirous." Imagine what a great accomplishment it is to bring up future servants of Hashem! This is what Chavah exclaimed when she bore Kayin, "Kanisi ish es Hashem," I have acquired a man - a son - not alone, but with Hashem! Giving birth to children and then raising them so that they will one day be servants of Hashem is an extremely important achievement for her neshamah, and it goes a long way in earning it the perfection that it needs. A woman is not expected to devote her hours to Torah learning, nor is she expected to meditate upon Torah ideas. She can certainly do so if she wishes, but her main area of achievement will be in transforming her neshamah through acts of chessed intended to benefit Hashem's people. A man, however, has to perfect his neshamah in a different way. While a man neither gives birth to children nor nurses babies, there are still multitudes of ways through which he can earn perfection. A man must go to work to earn a living, and there are very many nisyonos that he will encounter throughout his day. An entire section of Shulchan Aruch was written to deal with the laws that apply when one has to compete with another for a livelihood! There's a very great perfection a man can acquire when out making a living! Of course a man is also required to dedicate a part of his day to Torah learning, and other mitzvos that a woman wouldn’t have time to perform because of her day to day responsibilities. We see, therefore, that a man and woman secure perfection for their respective neshamahs in different ways.

Rabbi Avidgor Miller Speaks, pp. 245-246.