We consider the performance of riding is almost same. 

As for the difference, please view the drawing as below, it's an asymmetric rim laced by a rear hub.


As you can see the rear hub, the distance of two flanges to hub center is different. Left side (d1) is longer than right side(d2). So an asymmetric rim with offset spokes holes could balance the spoke tension force on two sides of a wheel. To balance the spoke tension, that's the advantage of asymmetric rim.

While, the spoke tension on two sides of an asymmetric rim gets balanced, the force in fact was transferred onto rim body, which makes rim left wall's positive force different from rim right wall's positive force. But for a standard (symmetric) rim, the positive force on two side walls of rim body keeps same.