jmc

Since $f$ has a degree of $2$, the degree of $(f(x){)}^3$ is $6$. Also, since $g$ has a degree of $3$, the degree of $(g(x){)}^2$ is $6$. Furthermore, as $f$ and $g$ both have a leading coefficient of $1$, then $(f(x){)}^3$ and $(g(x){)}^2$ both have a leading coefficient of $1$. Hence, when subtracting $(f(x){)}^3-(g(x){)}^2$, the leading terms cancel, and so $(f(x){)}^3-(g(x){)}^2$ has a maximum degree of $5$. We can see that a degree of $\boxed{5}$ can be achieved by taking $f(x)=x^2+x$ and $g(x)=x^3$, for example.