First, we find $g(4)$ by substituting $x=4$ into the expression for $g(x)$: $g(4)=3(4)+5=17$.
Next, we find $f(g(4))$ by substituting $x=17$ into the expression for $f(x)$: $f(g(4))=17+3=20$.
Similarly, we find $f(4)$ by substituting $x=4$ into the expression for $f(x)$: $f(4)=4+3=7$.
Finally, we find $g(f(4))$ by substituting $x=7$ into the expression for $g(x)$: $g(f(4))=3(7)+5=26$.
Putting it all together, $f(g(4))-g(f(4))=20-26=\boxed{-6}$.
The answer is: -6
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Let $f(x)=x+3$ and $g(x)=3x+5$.  Find $f(g(4))-g(f(4))$.

### Answer:
