The circuit in question is designed around N-channel MOSFETs. You are instead using NPN Bipolar Junction Transistors (BJTs).
These componentsAlthough both of these types of transistors can be used to switch power, they behave very differently from each other. One example is that MOSFETs don't have any significant gate current. A BJT, on the other hand, needs a current-limiting resistor placed before its base. Without this resistor, the current through the base can destroy the transistor (or the GPIO pin which is driving it).
Another issue with your specific transistor is that it has a 2.5V turn-on voltage (called \$V_{BE}(on)\$ in the datasheet). This is how much voltage needs to be applied to the base just to start current flowing through the transistor. Since the RPi outputs 3.3V (which isn't much higher than the threshold) the transistor may not "turn on" all the way. This may have compensated for the lack of current-limiting resistors!
Finally, when choosing a BJT (or MOSFET), make sure that its current capacity is sufficient. I don't know how much current your LED strips require, but make sure that you do the math :) The TIP120, incidentally, is good for 5A.
Short version: use an appropriate MOSFET instead of the TIP120's.