Does a Gentile have a soul? And what happens in the world to come?

As I understand it, the answer to your question is according to the great principle in the study of Jewish mysticism: that every level and tier is divided into five parts: nefesh, ruach, neshama, chaya, and yechida. Therefore, one must always specify and clarify regarding these five parts concerning which level is being discussed.

Now, it is brought in holy texts that there is an additional level in the very essence of the soul given to a Jew, more than the level included in the nefesh of a Gentile (see Tanya, Chapter 2) [and for this reason, the addition we received is called us "Banim"].

A Gentile has a nefesh behemith (animal soul) as explained in the Tanya. According to the aforementioned principle, we understand that this nefesh of the Gentile is also divided into the five mentioned parts, but they are all still contained within the "nefesh" part of him. In contrast, a Jew's "elokistic nefesh" [the additional part that a Gentile does not have] includes additional parts in the essence of ruach, neshama, chaya, and yechida.

As for what Rambam (Maimonides) wrote that the righteous among the nations have a share in the world to come, simply put, this means that their part and experience in the world to come corresponds to the root and essence of their nefesh. Since every Gentile in the world is connected to a Jewish soul (as the author of Tanya wrote in Chapter 37 that "he is connected to the life of one-sixtieth of the totality of the world that is dependent on his vital soul" - this totality of the world includes the part of the Jew in inanimate, vegetative, animal, and also speaking), similar to how the experience of a Jew's share in the world to come will be according to all the parts in its root that include nefesh, ruach, neshama, chaya, yechida, so too the Gentile's experience of their share in the world to come will be according to all the parts in their root, which are still within the "nefesh" part connected to the nefesh of a Jew.

Thus, it is not relevant to question "if they arrive in the world to come without a soul, then what is the difference between someone who arrives with a soul in the world to come and someone who arrives without a soul?" 

Because the intent in saying that a Gentile comes to the world to come without a soul means that all their experience of the world to come and their higher parts of chaya and yechida are still contained within the experience of the highest parts - inside of their nefesh. It is not relevant for them to experience more than the aspect of yechida within the part of the nefesh that is in them. In contrast, for a Jew, who comes to the world to come with an additional part of their actual neshama that a Gentile does not have at all, their experience of the world to come will be at a completely different level than the level of yechida within the nefesh experienced by the Gentile; for the Jewish soul has the ability to experience, with Hashem's help, even the aspect of the Ein Sof (Infinite) as it reveals itself in the very essence of yechida within them.

And this experience is not comparable to the experience of the yechida part that is only within the nefesh alone.

With blessings,

Rabbi Nachum