Here is how I remember this. I look at the sheet of charge from a long way away - so far, that I can't even tell how thin or thick it is. And I put my Gaussian pill box around the entire sheet.
When you have a non-conducting sheet, the charge density is "density through the entire volume". For a conducting sheet, you consider the charge to be divided between the two surfaces.
So if you have $\sigma$ on one side, and $\sigma$ on the other side, you have a total of $2\sigma$. But in the case of a non-conducting sheet, you just have $\sigma$.
After that, the two follow the same laws of physics...
An alternative explanation (that a Gaussian pillbox that extends on one side of the sheet only, and that sees half the charge but only has one surface with flux through it) results in the same outcome, and is physically more precise. But you asked for a "easy to remember" explanation.
There is a nice extended explanation including pictures at this site