I have 2 cylinders where I make a difference with boolean modifier. After applying, I see in the Edit mode that there are 2 edges connecting the outsite with the inside. Is there a way to avoid this or to join the separate faces?
If you want to convert that Ngon into quads delete it and use Ctrl+E > Bridge Edge Loops; it should work nicely if amount of vertices is the same.
However you can use more convenient way in order to construct cylinder like on your screenshot.
No, you cannot keep the Boolean Modifier from behaving like this ;-)