jmc

We apply Simon's Favorite Factoring Trick and note that if we subtract $30$ from both sides, then the left side can be factored. Thus, $$ab-6a+5b-30=343\to (a+5)(b-6)=343$$Since $a,b$ are positive integers, then $a+5,b-6$ must be a pair of factors of $343=7^3$, which are given by $\{a+5,b-6\}=\{1,343\},\{7,49\},\{49,7\}$, or $\{343,1\}$. Thus, $\{a,b\}=\{-4,349\},\{2,55\},\{44,13\}$, or $\{338,7\}$. Therefore the minimal value of $|a-b|$ is $|44-13|=\boxed{31}$.