jmc

The slope of a line through two points, $(x_1,y_1)$ and $(x_2,y_2)$, is $$\frac{y_2-y_1}{x_2-x_1}.$$Let $(x_1,y_1)=(4,5)$ and $(x_2,y_2)=(8,17)$. Then the slope of the line through the two points is $$\frac{y_2-y_1}{x_2-x_1}=\frac{17-5}{8-4}=\frac{12}{4}=3.$$Thus, $a=3$.

$b$ satisfies $y=3x+b$ for all points on its graph. Since $(4,5)$ lies on the the graph of $y=3x+5$, we can substitute $x=4$ and $y=5$ to solve for $b$. $5=3(4)+b$ and subtracting 12 from both sides yields $b=-7$. Therefore, $a-b=3-(-7)=\boxed{10}$.