To simplify notation define $\delta(x,y) = |x-y|$.
Here's an easier proof (using only Euclid) 
of the result in the answer that **Seva** posted:
a finite set of positive integers closed under $\delta$
a homogeneous arithmetic progression $\{d, 2d, \ldots, nd\}$.

Let $d = \gcd(A)$.  By Euclid's algorithm, $\gcd(x,y)$ can be obtained from
$x,y$ by repeated application of $\delta$, so $A$ is closed under 
pairwise $\gcd$; hence by induction $d \in A$.  Then $\max(A) = nd$
for some integer $n$, and $A \subseteq \{d, 2d, \ldots, nd\}$.
By repeated application of $\delta(\cdot,d)$
we find that $A$ also contains $(n-1)d$, $(n-2)d$, $(n-3)d$ etc.,
so $A \supseteq \{d, 2d, \ldots, nd\}$.  
Therefore $A = \{d, 2d, \ldots, nd\}$, **QED**.

It soon follows that if $A$ is an infinite set of positive integers
and $A$ is closed under $\delta$ then 
$A$ consists of all positive multiples of $\gcd(A)$.