Is there a relationship between the difference of the integers and the distance between them?

Well, there's really no such thing as the "distance between integers",
because numbers are not physical things that you can set on a table.

One way of representing integers is to draw a picture of a "number line",
and put marks at the positions of the integers.  When you do that, the
difference between two integers and the distance between their marks
on the number line are exactly the same.