Primes.

Note that since $x - y, y - z, x - z$ are positive, we must have $x > y > z$ .

We cannot have more than one of $x, y, z$ be even, since there is only one even prime. But if $x, y, z$ are all odd, then $x - y$ and $x - z$ are distinct even primes, a contradiction. Therefore exactly one of $x, y, z$ is an even prime, and since $2$ is the smallest prime, we must have $z = 2$ .

Thus $x$ and $y$ are both odd. But then $x-y$ is even and prime, so $x-y = 2$ and hence $x=y+2$ . Therefore our triple is $(y+2, y, 2)$ . This means that all $y+2, y, y-2$ are prime. But at least one of these is a multiple of $3$ , and the only multiple of $3$ that is prime is $3$ . The only possibility is $y-2=3\implies y=5, y+2=7$ .

Therefore our triple is $(x, y, z) = (7,5,2)$ .