A.R.Ty:
did you connect the -3,3 V for the contrast power correctly ?
And even more difficult, the display does not provide a negative power output. You will need an external negative voltage converter (see top of page 3).
Additionally the Backlight LED requires an external resistor. According to the datasheet:
Forward voltage Vf = 3.3 V
Assumed Operating Voltage V = 5V (from Arduino)
Max current: I = 45 mA
This is: R = (V - Vf) / I ~ 47 Ohm, so i suggest to connect the backlight via a 47 Ohm to the 5V of the Arduino. Well, it might be already damaged if it had been connected to 3.3V directly.
Most people (even if a little bit "electronical" ) don't know that there is a negative Voltage - me too for a long time.
But, as you can see, there is some.
Be happy that this one only needs -3,3V what is easy to build with some parts -most (usually the graphical ones) often use a negative voltage between -10 and -18 - THAT is a problem because for that you nee a complete other kind of technology to make it work.
The s.n. INVERTERS can power out some hundreds volts + or -, all by driving with i.e. 12V=: also they can invert the power from = to ~.
Now when you are more familiar with that look for displays WITHOUT needing negative voltage
fine that you got another LCD - its better to use those ones.
So see you again here when the 4x40 has arrvied 8) - but this is really easy to control (if you use a standard controller, best is always a HD44780 or compatible (this is another urgent "parameter" when buying Char LCDs))
) don't know that there is a negative Voltage - me too for a long time.