The main advantage is that with simmetrical power supplies (rails) you can handle AC signals without DC offsets. If you have only positive rail, the output signal can assume only positive values, in the other hand, with positive and negative rails, the output signal swings from negative to positive voltages.
For low power OpAmps, you can use the ICL7662 to generate the negative rail from a single positive supply (for example, you have 12V positive, ICL7662 generates -12V @ 50mA)