I have wired it exactly the same and used the exact same code for the program but my display shows nothing. I've heard that before.
The backlight works but nothing happens when I upload the sketch.The display should show 'something' even without running a sketch. This implies that your contrast setting (LCD pin 3) is incorrect so you must start there.
With pins 1,2,3,15, and 16 connected you should be able to vary the potentiometer and go from a blank screen (pin 3 = 5v) to a single row of blocks (pin 3 = 0v). If you can't do that then there is no sense trying to run a sketch.