smc

Using logarithmic rules, we see that $${\log }_{a}a-{\log }_{a}b+{\log }_{b}b-{\log }_{b}a=2-({\log }_{a}b+{\log }_{b}a)$$ $$=2-\left({\log }_{a}b+\frac{1}{{\log }_{a}b}\right)$$ Since $a$ and $b$ are both greater than $1$, using AM-GM gives that the term in parentheses must be at least $2$, so the largest possible values is $2-2=0\Rightarrow \boxed{0}.$ Note that the maximum occurs when $a=b$.