jmc

After converting both numbers to base 10, we subtract the values. We get $333_4=3\cdot 4^2+3\cdot 4^1+3\cdot 4^0=3(16)+3(4)+3(1)=48+12+3=63$, and $344_5=3\cdot 5^2+4\cdot 5^1+4\cdot 5^0=3(25)+4(5)+4(1)=75+20+4=99$. The difference is $63-99=\boxed{-36}$.