Explanation

x-y and x+y are both prime with the same parity (their sum 2x is even), so both are odd; hence x,y have opposite parity. Since x,y are prime and one is even, one equals 2. With y=2 and x an odd prime, x-2 and x+2 prime forces x=5 (3 and 7 prime; x=7 gives 9, x=13 gives 15, x=19 gives 21). x=2 is impossible since x-y>0 must be prime. Only (5,2) works: one combination.