We don't have assistant editors here so I can't say what the difference is.
With my company sub-editors do newspaper layout i.e fit stories and pictures in the spaces left by ads, check the stories, make sure they fit and make sense and write the headlines.
The editors here tell us what stories and pics to put in the first place (and we often have to tell them they don't fit).
In newspapers I guess assistant editors help editors... maybe on larger papers?