I like Grijesh's answer, simple and elegant. Here is another take on it, using a recursive call:
global sum
def sum_between(a, b):
    global sum
    # base case
    if (a + 1) == b:
        return sum
    else:
        sum += (a + 1)
        return sum_between(a + 1, b)
Not as straight forward as using sum(range(a+1, b)). But simply interesting as an exercise in recursive functions.