Existence of two primes satisfying the given conditions I want to know whether 
the equation 
$x^a-x=y^b-y$
has a solution or not satisfying the conditions that $x$ and $y$ are distinct odd primes, $a$ and $b$ are integers both greater than $1$.
 A: $$13^3-13=3^7-3=2184$$
I should say how I found this.  It came from looking at Section D9 ("Catalan conjecture") in Richard Guy's magnificent Unsolved Problems in Number Theory (third edition, pg. 238), where he writes:

"Leech asks if there are any solutions of $|a^m-b^n|\lt|a-b|$ with
  $m,n\ge3$.  With equality he notes $|5^3-2^7|=5-2$ and
  $|13^3-3^7|=13-3$.  Are these all? are the shared exponents $3$, $7$
  significant?"

Unfortunately there's no reference given for Leech.
Its presence in UPNT suggest the general problem of finding integer solutions to $x^a-y^b=x-y$ with $a\gt b\gt1$ is difficult.  The OP's restriction to (odd) primes $x$ and $y$ might make things easier, or it might not.  In any event, there is at least one solution; if there are others, my guess is they're likely to be hard to find.
A: Michael Bennett, On some exponential equations of S. S. Pillai, Canadian Journal of Math. 53 (2001) 897-922, reports eight solutions of $x^a-x=y^b-y$ in positive integers, with $a,b>1$:  
$6=2^3-2=3^2-3$,
$30=2^5-2=6^2-6$,
$210=6^3-6=15^2-15$,
$240=3^5-3=16^2-16$,
$2184=3^7-3=13^3-13$,
$8190=2^{13}-2=91^2-91$,
$78120=5^7-5=280^2-280$,
$24299970=30^5-30=4930^2-4930$. 
See the OEIS for links. I don't think any new ones have been discovered, and I don't think anyone has proved there are none. Of those eight solutions, the only one satisfying the (rather unmotivated) primality stipulations is the one Barry reported in his answer five years ago. 
