The two power buttons do carry out the same function of powering up your system but in various ways.
The first button to the right-hand side is the regular power button for starting your system normally (either Windows or other operating system).
The second one to the left-hand side is called the 'QuickStart' button. It is used to power up the system into the ASUS 'Expressgate' which is meant for Linux operating systems. If you aren't having a Linux OS like Ubuntu or Mint in your computer this button would be of no great use to you.