Julia, 69 bytes
This reads a space-delimited pair of integers from STDIN and prints the result to STDOUT. The only thing I've really golfed here is whitespace; otherwise this is probably how I would go about it in a non-golfing context.
Ungolfed + explanation:
# Read a line from STDIN, split it into an array on the space, convert
# the elements to integers, and assign the first element to a and the
# second to b
a, b = int(split(readline()))
# Get the primes between a and b inclusive. primes(x) returns the primes
# <= x, so the set difference of primes(b) and primes(a) will get us only
# those between a and b
d = setdiff(primes(b), primes(a))
# Print the sum to STDOUT
Just realized how old this challenge is.