As UnknownVT and others have pointed out, a lot of the problem is due to the way our brain processes visual input. We internally declare any of a wide range of spectra to be "white" after being exposed to them for even a fairly short period. I first became aware of this a long time ago when we had chosen a very slightly off-white color to paint a room -- the paint mixer added just one glurp of green to a can of white. I was painting in a ceiling corner, where I was surrounded by the color. After some time, I stepped down, turned around -- and everything except the wall had a distinct pinkish hue. My brain had decided that the wall was "white", so everything else was pinkish. This is happening to me a lot now with the various lights in my house.
If you shine a white (say, hot incandescent) light through an object, say a thin piece of fabric, the object absorbs some wavelengths more than others, resulting in what we perceive as colored light. The spectrum that gets through probably has a bunch of lines of various wavelengths and intensities, and ranges of wavelengths where there isn't much being reflected. As it turns out, you can make a light consisting of just three spectral lines and, by adjusting only their amplitudes, have it look like exactly the same color as the complex spectrum from the filtered white light. The spectra are very, very different, yet they look just the same to us. This is, of course, the basis of color television, which produces nearly any color by combining just three. (As I recall from long ago, you can't quite imitate all possible colors, but you can get very nearly all by careful choice of your three primary colors. It seems our eyes and brains can be fooled pretty easily.
The problem is that if you shine the two apparently identically colored lights on something of another color, you'll get different results. If, for example the three-color light is made of pure red, green, and blue, and the object reflects none of those exact wavelengths, it'll look black. The filtered light might contain some of the wavelengths the object reflects, so the object will be some color and not black. This happens with "white" lights, particularly so with fluorescents which tend to have spectra consisting of a number of discrete lines rather than a more continuous, filled-in spectrum. And that's why an RGB LED light might look white (or any other color we want by adjusting the three LED intensities) but won't render colors anything like a heated filament (black body) light.
c_c