Solution:

The numbers $A - B$ and $A + B$ differ by $2B,$ so they have the same parity. Being prime, they must both be odd, which forces $A$ and $B$ to have opposite parity.

Since $2$ is the only even prime, $B = 2.$ Then $A - 2,$ $A,$ $A + 2$ are three primes forming an arithmetic progression of odd numbers, which must be $3, 5, 7.$

The four primes are $2, 3, 5, 7,$ and their sum is $17,$ which is prime.

Thus, the correct answer is E.