Solution:
We claim that $(8, 2, 3)$ is the desired solution.

Observe that $a^{c-b} = b^{c}$, so clearly $a \neq 1$ and $b < a$. Furthermore, $a$ and $b$ must be distinct powers of the same integer.

If $a$ and $b$ were powers of an integer $n > 2$, then we would have $a + b + c \geq 3^{2} + 3 + 1 = 13$. Thus, we only need to consider when they are powers of $2$.

If $(a, b) = (4, 2)$ then $(c-b) = \frac{c}{2}$, so $c = 4$, which makes the values not distinct.

If $(a, b) = (8, 2)$ we get our aforementioned solution.

Any other $(a, b)$ sum to at least $12$, in which case $a + b + c \geq 13$.

Thus $13$ is minimal.