It's a great question.
If you want to differentiate them, you should understand the words at first——words carry meaning.
The original meaning of 边 is a side of a geometrical figure: consider a triangle ABC. AB, BC, or CA are 边.
As you can see the AB, BC, CA are the borders of the triangle, there is a meaning representing edges, banks, or remote regions.
And 面 carries the meaning of "face". Your face is an approximate 2D figure, though strictly it is not. Consider a pyramid ABCDE with summit A on top, we will call ABC, ACD, ADE as 面.
However, as you know, a line cuts 2 sides in 2D space, so does 面 separates 2 space in 3D one.
We can say 一边... 一边... to express things happen simultaneously.
He sang while he worked. 他一边(面)唱歌一边(面)工作