Difference between a Power Plane and a Copper Fill?
I don't know Eagle, but in other tools the difference has to do with how the gerbers are generated.

For a power plane, the gerbers will be generated as a negative image. The gerber file will indicate features for regions or shapes where copper should be removed. 

For signal layers, the gerbers will indicte features for shapes where copper should be retained (not etched away). 

This can have a big effect on the size of the gerber files. For a polygon on a signal layer, the gerber will usually include a long self-overlapping serpentine trace (or mesh of crossing traces) to fill the polygon area. This can lead to a very large gerber file. 

In the days when gerbers were transferred to vendors over dial-up modems, it made a much bigger difference to reduce the size of the files.

There can also be a subtle difference in the actual generated boards, because the features (whether positive features in a polygon, or negative features on a plane layer) must be generated by actual apertures on a photoplotter tool, selected from a limited set of available shapes and sizes.