The days of Moshiach and olam habbah (the world to come) are two separate things (Sanhedrin 99a). There is a controversy whether the days of Moshiach will be of the natural order or of a miraculous nature. But there is uniformity that people will have physical bodies.
Ramban and Rambam have a fundamental disagreement regarding Olam Habbah. Both still compare our nature there to that of angels, but they yet disagree. Rambam says it is a soul experience without a body. Ramban disagrees. Rambam further holds that a person, after death, goes to olam habbah of the souls. Ramban says that the place after death (without a body) is not on the same level as olam habbah in the end of times.
The Gemara Sanhedrin 90a considers the revival of the dead as an aspect of olam habbah. The Rambam explains that it is not the final stage, but after the revival of the dead into physical bodies (like in the days of Moshiach), the person again dies for olam habbah. Ramban says the revival of the dead is for the purpose of everlasting life, not to die again.
Rambam’s position is difficult both in comparing the revival of the dead to the times of Moshiach and because Daniel 12:2 states about the revival of the dead that it is for eternal life.
Rambam proves from the Talmud (Berachot 17a) that olam habbah is a spiritual experience. Ramban answers this explaining that the body of the revival goes through a transformation process becoming a soul kind of body which does not do physical things. Raavad and Ramchal also follows this approach.
The soul itself has a form. There are numerous souls and each has some sort of location and form. Only God is without any form. Ramban’s soul body is a second form of a body that is related to the regular body but does not impede on the soul. Ramchal says that the such body is very minute. Rambam considers such a body to be a tag-along without purpose or reason. But if there is indeed some remainder of it, there is a reason and a benefit.
On the other hand, Rambam’s view can also work within the framework of Ramban. After the revival of the dead, the body goes through a transformation process. This is not death to olam habbah, but a transformation process of everlasting life. Some aspect of the body might remain (like Ramban), but it can also be said that the body transforms completely into the soul (like Rambam). Even so, there is a difference between after death and the end of times because the soul does not enjoy seeing the body rot in the earth.