White Light is Just an Illusion
Pretty much any light source which is emitting "white" light has a different spectrum. There are a few reasons: blackbody radiation and the method of producing light. When our eyes see radiation of several different wavelengths at once, we can combine them and make them "white" or "pink" or whatever color it is.
Filament bulbs (and stars!) produce light via blackbody radiation. It just so happens the curve for these things fits into the visible spectrum well, and we register it as "white" light. This is, however, why filament bulbs have that slight yellow quality.
Light of Varying Wavelengths Team Up to Make White Light
If you take the spectrum of different lamps, you'll see they are composed of different combinations of wavelengths. They all produce what is called "white" light, but they each have different wavelengths of varying magnitudes. This results in different light bulbs having different qualities of "whiteness" or "color" to them, and is something interior designers worry about. ("Yes, that fabric looks good now, but what about in artificial light?")
White LEDs are Hardly LEDs at All!
For instance, that LED isn't a "white" LED. It's a blue one which drives a filament. The actual LED is why you get the nice peak in the blue area, but then the filament is producing that "blackbody radiation" curve after it. It's why white light from an LED looks blueish; it is emitting more blue light than other light. A true white LED would have three spikes in its emission spectrum, and would likely be composed of three different LEDs.