GaN provides a couple of benefits compared to Silicon:

1. It offers a lower on-resistance for the same breakdown voltage as a silicon device [1].

2. It offers a lower gate charge which allows you to turn it on and off at a much higher frequency than Silicon devices [2]

(1) means you have lower losses in a power circuit with GaN vs Si (leading to more compact - smaller heat sinks - power supplies for example)

(2) means power supplies can run at much higher frequencies. At higher frequencies, you can use smaller inductors and capacitors which means more compact circuits.

A practical example of GaN in action is FinSix' Dart power supply [3]

[1] Slide 5 of http://www.irf.com/product-info/ganpowir/gangeneral.pdf [2] http://m.eet.com/media/1179793/si-vs-gan-vs-sic-which-proces... [3] http://finsix.com/dart/