Growth is number of people per unit time. Rate of growth is number of people per unit time squared. Just as with speed vs. acceleration, there certainly is a distinction.
I'm not a native speaker so I could be wrong. But in the wikipedia article about exponential growth, "growth rate" is defined as the first derivative with respect to time, which contradicts what you are saying: https://en.wikipedia.org/wiki/Exponential_growth#Differentia...
I understand the distinction you are trying to make, but I think what you call "growth rate" would actually be "growth acceleration".