Well, basically the the live rear axle links the left and right wheels together with a solid beam, meaning that movement on one wheel affects the other. In an independent suspension, the drive wheels can move up and down independently, without one affecting the other.
The main disadvantages to LRA is the tendency to lose traction when cornering and as you said a stiffer and harsher ride on rougher roads.
Fans would argue that LRA has an advantage in drag racing due to its stiffer design. But as far as I know that's about it. Independent suspension has the advantage in about everything else.
Richard Hammond from Britain's Top Gear once described the LRA as "running around with your laces tied together, there are better ways of getting around"