It's a little hard to tell on my screen, but it looks like you may only have preview turned on in view port (i.e. you selected the object and pressed "2" on the keyboad) instead of actually smoothing the mesh...the little blue line at the edge of the cube is what's suggesting this. In short, Maya only renders the physical mesh and not the preview...preview doesn't actually alter the mesh, it just gives you an idea of what it will look like once it's been smoothed. Try this...
1.) Save a copy of the file...just in case I'm wrong (always, always, ALWAYS save back up copies of your files!)
2.) Click on the mesh, then press 1 on your keyboard...you should see some change in how the mesh looks.
3.) In your polygons menu set, with the object still selected, go under "Mesh" and click "Smooth", then try re-rendering and see if that's closer to what you had in mind.
Good luck!
