Hi Jason,
First of all thank you for providing the file.
I think the problem is the expectation you have. The two Meshes have Vertices in different positions and, if you. for instance try to Shrinkwrap a Cube to an Icosphere, the 8 Vertices of the Cube snap to the Surface (or Vertex, depending on the Mode) of the Icosphere:

Using a Subdivision Surface Modifier after that Shrinkwrap, will make the Cube more rounded, but not make it more like an Icosphere.
As the two Plane Meshes are already closer, the difference will be less extreme, but they can never become exactly the same with this method.