The Arduino uses a linear regulator, and with linear regulators any voltage above the regulated voltage is wasted as heat. There is no advantage to using a higher voltage power supply, battery or otherwise.
9V batteries have about a 600mah capacity. Generally just a poor choice of battery given their cost.
Also, batteries are typically rated for a 10 hour discharge. If you pull 60ma from a 9V battery you'll probably get around 600mah capacity, but if you pull 200ma you'll only get about a 500mah capacity. The faster you drain the less total amp hours you get.