Both Rene Descartes and Plato believe that we house something incorporeal; Descartes calls it the mind, while Plato calls it the soul. They also both believe that when we die, although our physical body is gone, the soul/mind continues. It is unclear where the soul/mind goes after the body dies, but both philosophers believe that, as an incorporeal part of us, it cannot "die" like our physical body. They are also both described as dualists, believing that the physical body and intangible soul are different and distinct from one another and are not tied together; they both believe that when the body dies, the soul does not die with it.
Despite these similarities, they differ on a few points. Descartes works his theories around the Church and Christianity, so his philosophy involves the concept of heaven and judgement. Descartes believes that the soul goes to the heavens, but the mind is immortal. Descartes's mind does not need the body, and it is therefore immortal. Descartes also believes that the soul is joined to the body. He wrote that
We need to recognize that the soul is really joined to the whole body, and that we cannot properly say that it exists in any one part of the body to the exclusion of the others. For the body is a unity which is in a sense indivisible because of the arrangement of its organs, these being so related to one another that the removal of any one of them renders the whole body defective. And the soul is of such a nature that it has no relation to extension, or to the dimensions or other properties of the matter of which the body is composed: it is related solely to the whole assemblage of the body’s organs. This is obvious from our inability to conceive of a half or a third of a soul, or of the extension which a soul occupies. Nor does the soul become any smaller if we cut off some part of the body, but it becomes completely separate from the body when we break up the assemblage of the body’s organs.
Thus, Descartes believes that the soul is tied to our body and is only freed one the body dies or the organs are no longer present.
Plato also believes that the body and soul are distinctly different entities. Plato is described as a dualist, believing that when the body dies, the soul lives on. This is similar to Descartes, although Descartes believes it is the mind that lives on as an immortal concept. For Plato, the soul becomes free again once we die, before being trapped again in another body. Plato also believes that the soul is responsible for a person's mental or psychological activities.