Led driver - constant current or constant voltage
Your driver has an output of 416 mA @ 12V and is constant current (CC).

You need to know the voltage input to the LED strip. 

Based on this picture it appears there is one current limiting resistor for each three LEDS. What make this more difficult is, in the photo, the ends of the strips are cut off, you say there are 32 LEDs, and the PCB says 36 LED.  Three LEDs per resistor makes more sense if there are 36 LEDs (12 resistors) than 32 (10.666 resistors).  I cannot see how the strips are wired.   



The five needed unknowns


What is the voltage of the driver output when connected to a strip?
What are the values of the resistors?  
How many resistors?
Are there 32 or 36 LEDs per strip?
Are the strips wired to the driver in series or parallel?


I am going to assume the power supply should be 12V constant voltage.  Although a CC driver may (or may not, depends on the value of the resistors) work with one strip.  If the strips are wired in series, the pictured driver would not work becasue you would then need 24V. 

If the driver does not light up one strip by itself then I would guess the board needs more than 416 mA.  5 watts seems to be very inadequate for even one strip with 36 LEDs. They appear to be high power LEDs like a Cree Xlamp with a 350 mA test current. So the strip may be trying to draw more the 416 mA which drops the driver voltage below 12V and the LEDs do not have enough voltage to light up.  

I do not know what RS means.

UPDATE

The 241 on the resistor mean 24 x 101 or 240Ω
So the current flowing through each resistor is about 37 mA 
Which means the 3 LEDs and resistor = 444 mW
.444 W x 36 resistors = 5.328 W x 2 strips = 10.656 W

You need a plain old power supply, not an LED driver.
Power Supply Needed: 12VDC out and at least 1 Amp or 12 Watts.
RS 12V Power supplies