What is the difference between the greatest and least possible areas of the rectangle

You haven't given any information at all to describe the rectangle ... perimeter, dimensions, etc.  That's kind of important.

Using all of the information you provided, I can only tell you this:

-- The greatest possible area = (1/4 of the rectangle's perimeter)² .

-- The least possible area = the smallest positive number you can think of.

-- Their difference is

   (1/4 of the perimeter)² - (the smallest positive number you can think of) .

The answer would be 7 cages