For the gamma distribution Mean (Mu) = aB and variance (sigma squared) = aB^2
For the exponential distribution Mean (Mu) = B and variance (sigma squared) = B^2
Examine these and you can see the difference.. The means are basically the same except gamma has a and exponential does not. Variance for gamma also includes a, while exponential does not.

If a = 0 and B = 0 then Mu for exponential is 0 and sigma^2 = 0.
If a = 0 and B = B then Mu for exponential is B and sigma^2 = B^2
If a = 1 and B = B then Mu for exponential is B and sigma^2 = B^2, but note that the Gamma Distribution will also have Mu = B and sigma^2 = B^2.

That's the key, the exponential distribution is EQUAL to the Gamma distribution when a = 1 and B = B.