Shortcomings of the PMS* theory of souls:
1 - PMS doesn't explain how the soul and body interact. PMS says that the soul can influence our actions, and that our actions can improve our souls. But it provides no mechanism which explains how this occurs.
2 - PMS invites abuse from people who prefer to focus on the next world, rather than this world. For instance, some people imagine that so long as they pray or keep shabbos their ticket to the next world is punched no matter how they treat their fellow men.
3 - PMS is embraced by people who are looking for shortcuts. "Why bother busying myself with the hard work of studing and acting ethically," some seem to say, "when my soul can also be elevated by eating kugel or listening to music, or by the rote recitation of tehillim or other prayers?"
Shortcomings of MTS (my minimized theory of souls):
1 - MTS doesn't explain how the soul is improved. We believe that God rewards or punishes the soul after we die, but in order for this final judgement to have any meaning, our actions in this world need to have some influence on our souls. For a time, I thought the Rambam offered a solution, but I was wrong. The Rambam teaches that the soul is nothing but our developed intellect (He holds that those of us with undeveloped intellects aren't punished at death, per say; rather we cease to exist.) This view explains how the soul is improved through our actions, but it doesn't tell us how the developed intellect achieves immortality. I also thought epiphenomenalism might get around this problem; but epiphenomenalism only explain how the body influences the soul without the soul influencng the body, we're still left with the question about how immortality is achieved.
2 - MTS doesn't explain how free will operates. If the soul isn't influencing our thoughts and actions, how can we avoid saying that every event (human thoughts and actions included) is causally determined by an unbroken chain of prior occurrences? This denies free will. And though many Jewish thinkers (notably Rav Y.B Solivetchik and Eliyahu Dessler) argue that a great deal of out thoughts and actions are the product of circumstances, not free will, I can't find anyone (save Hasdai Crescas and some Chabadniks) who advances a Jewish doctrine of predestination. Neither the Rambam nor epiphenomenalism offer any help here: The Rambam held that the developed intelect/soul influences our actions and epiphenomenalists deny free will. Compatabilism solves the problem but only by redefining free-will; under compatabilism our actions aren't the product of an actual choice, but a reflection of our characters. This falls short of the Rabbi's definition of free will.
Ultimately choosing one theory over the other is to express a preference for one mystery over another. PMS can't tell you how the soul and body interact, neither can MTS. If anything recommends one view over another, it is this: MTS leaves less to faith, and it provides a world-view that de-emphasizes shortcuts to spiritual growth.
We'll give it to Aristotle who wrote that "To attain any assured knowledge of the soul is one of the most difficult things in the world." No kidding.
*The PMS (Popular Meaning of Souls) theory holds that souls have their own agency: They make choices and decisions, they have desires and needs, they can affect what happens in our brains, and they can be inspired or otherwise affected by kugel, music and so on. MTS (my Minimized Theory of Souls) says most of that can't be proven, and therefore should be rejected.