There are a number of ways you can smooth a polygon mesh in Maya, each offering their own advantages and disadvantages.
Smooth mesh preview quickly shows you how a polygon mesh will appear when smoothed, without actually smoothing the mesh.
This mode is ideal for tweaking your mesh prior to an actual smooth mesh operation, to ensure the end result looks as close to your desired result as possible. You can also use it to smooth without recording any history and allows you to make changes to the components on the smooth mesh preview itself without the cage necessarily being visible.
For more information, see Preview a smoothed polygonal mesh.
Unlike the smooth mesh preview, a subdiv proxy actually creates a smoothed mesh and places it inside the original mesh. These two meshes are linked via their construction history. Any edits you make to the original mesh are reflected on the smoothed version underneath.
In most cases, a smooth mesh preview is superior to a subdiv proxy. However there are a few times when you would want to use a subdiv proxy, which are:
For more information, see Work with polygon meshes using Subdiv Proxy.
A subdivision surface allows you to subdivide specific regions of a mesh. This gives you the ability finely tune or smooth certain areas without changing the entire mesh.
Subdivision surfaces help you keep a lower geometry count since they only insert control points where you need them (instead of uniformly across the mesh). However, it is also has the slowest performance of the three smoothing methods. Subdivision surfaces are largely unsupported in pipelines outside of Maya, meaning you have to convert them to polygons or NURBS before exporting to these pipelines.
For more information, look for the topic 'What are subdivision surfaces' in the Subdivision Surface Modeling guide in the Maya documentation archive, available at http://www.autodesk.com/maya-docs-archive.
In Maya, the polygon smooth feature (Mesh > Smooth) works by adding divisions to the polygon mesh. However, the smooth feature in Autodesk 3ds Max works by averaging existing components. This is much closer to Maya’s Average Vertices (Mesh > Average Vertices) feature.
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