Plato thought there was an ideal world where ideas live. Independently of people. These ideas throw a shadow on the material world we live in. By examining this material world we can get knowledge of the world of ideas. But this knowledge will never be perfect. We can make a 3D cube from stone or we can construct an image of it in the head but it will still be a reflection or shadow. Math gets close but still it doesn't give the idea itself. Maybe Plato had a different view regarding the last (his cubes, tetraheders, dodecaeders, etc. are thought by him to be perfect "Platonc" objects corresponding to how the ideal world looks like). I am not sure. Of course there are many more objects thinkable than his Platonic objects.
Is his ideal world comparable to the concept of heaven? Of course the question is what I mean by comparable. Well, both offer an ideal world. An idealized version of reality. In heaven there is no wordly suffering, for example. In the ideal world of Plato the ideas don't suffer from human contamination. What are the differences? Can his ideal world be compared with heaven in the first place?