jmc

The largest power of $5$ that divides both terms is $5^3$. We factor out $5^3$ as follows: $$\begin{array}{rl}{5}^5-5^3& =5^3\cdot 5^2-5^3\cdot 1\\ & =5^3(5^2-1)\end{array}$$ $5^2-1=24$, which factors as $2^3\cdot 3$. Thus, the prime factorization is $2^3\cdot 3\cdot 5^3$, and the sum of the prime factors is $2+3+5=\boxed{10}$.