The surface area of a cylinder is given by the formula $2\pi r^2 + 2\pi rh$, where $r$ is the radius and $h$ is the height. Substituting $r=4$ and $h=7$, we have $2\pi (4)^2 + 2\pi(4)(7) = 32\pi + 56\pi = 88\pi$. To round this to the nearest thousandth, we find approximately $88\pi \approx \boxed{276.460}$.
