I realize that both msh210 and IsraelReader already gave answers based on this source, but I think that neither one explained it fully/properly, so I would like to attempt to answer it myself.
This question was asked to R. Avraham the son of Rambam. This was his answer:
Birchas Avraham # 44
ואותה הקושיא שהקשית על עיקר דבריו לא קשיא עליה שבמנין המצוות בתחלת
הלכה אמר שהמצוה לישא אשה בכתובה וקידושין ולא אמר לקדש אשה וכן אמרינן
לישא דקא עבד מצוה וזה שאמר וליקוחין אלו מצות עשה לפי שהיא תחלת מצות
הנשואין וכך אמר בתחלה יקנה אותה תחלה בפני עדים ואחר כך תהיה לו לאשה
שנאמר כי יקח איש אשה ובא אליה אבל קידושין בלא נישואין ודאי לא השלים
המצוה עדיין ומצות פריה ורביה מצוה אחרת היא שמצות פריה ורביה כשיהיה לו
בן או בת קיים המצוה ומצות הקדושין והנשואין אפילו יש לו כמה בנים וכמה
בנות ויש עמו כמה נשים כל אשה שירצה לישא אותה מצוה עליו שישא אותה
בקדושין ועיין בדבריו ז"ל בספר מצות יתבאר לך ההפרש שביניהם ואין בנו
צורך להאריך בדבר זה
My understanding of this is as follows: Rambam holds that there is a mitzvah of marriage which is made up of two components. 1. Kiddushin 2. Nissuin. If one performs Kiddushin but never gets around to doing Nissuin, he has started the mitzvah but has not completed it, and thus he is not considered to have fulfilled the mitzvah.
Therefore, the Talmud in Moed Katan means that when one does Nissuin he is fulfilling a mitzvah (because having already done Kiddushin, he is now completing the mitzvah) but when one does Kiddushin he is not fulfilling a mitzvah (because not having done Nissuin, the mtzvah is incomplete). This explains how Rambam's view is to be reconciled with the Talmud.
As for how, R. Avraham fits this explanation into Rambam's words (or how he derives it from Rambam's words), I believe the following is correct:
There is no indication of this explanation from Sefer Hamitzvot. There Rambam lists two mitzvot, one about procreation (#212) and one about marriage (#213). It is certainly implied that these are two separate mitzvot (i.e. one can fulfill the mitzvah of marriage without procreating), but there is no indication here that the mitzvah of marriage consists of Kiddushin + Nissuin. If anything, the implication there is that Kiddushin is the entirety of the mitzvah of marriage, as that is all that is mentioned there.
However, the indication that Rambam's view is in fact that only Kiddushin + Nissuin = the mitzvah of marriage, is in Mishneh Torah. In the beginning of Hilchot Ishut Rambam lists the mitzvot that are included in this section. The first one is:
לישא אשה בכתובה וקידושין
To marry a woman, granting her the rights of the formal marriage
contract (ketubah) and sanctifying the relationship through the rites
of kiddushin; (Chabad.org)
R. Avraham picks up on the fact that Rambam does not say simply "לקדש אשה" which would be sufficient if Kiddushin was the entirety of the mitzvah, but instead says "לישא אשה בכתובה וקידושין" which indicates that there is an additional component, namely, Nissuin.
R. Avraham does not note this, but in Shemonah Perakim (Chapter Eight) Rambam has a similar formulation:
לפי שהאשה הזו אם לקחה בכתובה וקדושין והיא מותרת לו ונשאה לפריה
ורביה הרי זו מצוה (Kafih translation, my emphasis)
R Avraham then clarifies that the mitzvah of marriage (i.e. Kiddushin + Nissuin) is distinct from the mitzvah of procreation. The mitzvah of procreation is fulfilled by having a surviving son or daughter. The mitzvah of marriage can be fulfilled even if you already have many sons, many daughters, and many wives. Any woman that you want to marry, there is a mitzvah to marry her via Kiddushin = Nissuin.
