When playing a piano piece, is playing it in E "minor" just a different way of saying it is being played in E "flat"?

No, minor and flat are not the same thing.  

Do you mean Eb major or Eb minor? The answer is no in either case. The E minor scale begins on E and the Eb scale begins on Eb. That's two different notes.

No it is not.  The scales for E minor and Eb are very different.

E and Eb are different notes. Minor and Major are different modes. Both E and Eb can be major and minor.

People often think of minor as being "lower" than major, perhaps because the interval of a minor third is smaller than that of a major third. 

No, the keys of Em and Eb are completely different things!