First, we find $g(4)$ by plugging $x=4$ into the equation $g(x)=3x+5$.
So, $g(4)=3(4)+5=17$.

Next, we find $f(4)$ by plugging $x=4$ into the equation $f(x)=x+3$.
So, $f(4)=4+3=7$.

Now, we can calculate $f(g(4))$ by plugging $g(4)=17$ into the equation $f(x)=x+3$.
So, $f(g(4))=f(17)=17+3=20$.

Similarly, we can calculate $g(f(4))$ by plugging $f(4)=7$ into the equation $g(x)=3x+5$.
So, $g(f(4))=g(7)=3(7)+5=26$.

Finally, we can calculate $f(g(4))-g(f(4))$.
So, $f(g(4))-g(f(4))=20-26=\boxed{-6}$.

The answer is: -6
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Calculate the value of $f(g(4))-g(f(4))$, where $f(x)=x+3$ and $g(x)=3x+5$.

### Answer:
