RGB LED brightness levels
The red, green and blue of RGB LEDs do have different perceived luminous intensities, for several reasons. Some of them are listed below, not in the order of importance:


Eye color sensitivity 


The human eye is sensitive to different colors to different levels - and this relationship also varies by light intensity: Both ambient light intensity (bright room versus dark room) and emission intensity (bright LED versus dim LED).
Under photopic vision conditions i.e. well lit conditions where the cones of the eye are the primary mediators, we are typically most sensitive to green light, approximately 555 nanometer wavelength

(source)
Under scotopic (dimly lit) conditions, this drops down to around 507 nm aqua color as the most sensitive. 
These sensitivity patterns change with age, differ by gender, and vary from person to person. Also, traumas, including even purely psychological traumas, can change the vision profile.

LED power emission


This one is easy: Let us take this RGB LED datasheet as an example.
Before factoring in the efficiency differences in conversion of electrical energy to light in different LED junction chemistries, the power expressed across a 2.0 Volt junction (typical Vf for red in the example LED) at 20 mA = 40 mW is clearly different from the power across a 3.2 Volt junction (for blue and green) at 20 mA = 64 mW.
Hence, the amount of light emitted by each junction at the same current should be different: Not proportionally so, since efficiencies are vastly different, but different nevertheless.

Color matching between LED junctions:


This is actually a farily complex task in itself, even if all else is ideal. Some useful links to give you an insight into exactly how complex it is to match R, G and B in LEDs: 
This article computes a ratio of estimated R:G:B current values of 2.77:5.79:1 or around 3:6:1, to achieve an evenly matched white, using one specific RGB LED combination. Your results will of course be very different due to using a different LED.
This write-up touches upon some of the hoops one has to jump through, to achieve an even RGB output (again, to get a white light) and to then sustain the matching as the LEDs age and lose luminosity at different rates.  

One could go on listing other factors...

Specific to the question: Current through each LED


The VCE(sat) of the TIP120 at 20 mA is approximately 0.75 Volts (Figure 2 in datasheet). Thus available voltage for the LEDs and any current limiting resistors should be 5.0 - 0.75 = 4.25 Volts
Assuming that the supply rail is a solid 5 Volts with unlimited current capacity and does not droop under load, the LEDs should all have blown out when operated without current limiting resistors, under these conditions. That didn't happen, which means some other current limiting is at play. 
Either the 5 Volt rail from the RPi cannot sustain the current requirement of say 60 mA for rated operation of all 3 LED junctions (unlikely, but possible), or there is some other resistance coming into play: Long thin wires? Long USB cable supplying the RPi? RPi drawing close to the maximum available current already? Too low a base current on the TIP120? Only you can tell, by some judicious poling around with a multimeter.
Any which way, if the supply, at the LED anodes, is unable to sustain the current needed for the 3 LEDs, and is therefore dropping in voltage, this drop will fall below 3.2 Volts before it drops below 2.0 Volts. At something around 3.0 Volts, still working on the example LED datasheet, the green and blue LEDs will see reduced conduction, tailing off to non-conducting - so the demanded current will reduce. Somewhere in that vicinity, the supply rail will reach an equilibrium with barely illuminated green and blue, and yet there will be plenty of voltage headroom for the red LED to light up. 
This matches with our observed behavior.
Either that, or the supply voltage from the RPi is not 5 Volts as supposed, but actually 3.3 Volts, which will again give exactly the observed behavior: A multimeter would be able to confirm this. 





Now, about resistance calculation for the LEDs:

Taking the typical voltages from the datasheet, for 20 mA current each, and supply voltage of 4.25 Volts calculated previously:


Rred = 112.5 Ohms, ~ 120 Ohms
Rgreen = 52.5 Ohms, ~ 56 Ohms
Rblue = 52.5 Ohms, ~ 56 Ohms


Those 3 values should give approximately 20 mA current through each color junction, if the supply is truly 5 Volts.