Apologies if this is a stupid question but I haven't been able to figure it out. In both cases the images are adjusted such that the color aren't influenced by the color of the illumination right? Or am I missing something?
Color calibration usually refers to adjusting a device, such as a monitor or printer, so that digital colors appear in a standard way. An image should look as close as possible to the same across different displays and media, and calibration is a necessary step for that.
White balance, as you say, is adjusting an image so that the illuminant appears neutral (or, if not neutral, to a color temperature shift from neutral as desired by the artist).
So the improved real world color accuracy is just a byproduct of color calibration? The intent is not to match the real world colors but to ensure that the colors appear the same between devices?
There is no
spoon real world, only perception. A white piece of paper stapled to a tree will look different to you based on time of day, illuminated by a street lamp or flashlight, or the moon. You've lived enough in this world to where you have developed a sense of what's natural and what is not.
So, when photographing that paper, you get to decide what the white balance is and shift the captured scene anywhere along the warm-cool axis. It's completely subjective. That's not to say that all choices will appear natural - they won't. But, it is to say that you ultimately have control over the color temperature of your shot, and that control is called white balance.
the images are adjusted such that the color aren't influenced by the color of the illumination right?
This is not always true. The color of illumination is sometimes desired. Warm morning light or incandescent bulbs for example. Or, to your point, we'd want to correct for a sterile feeling fluorescent. Or maybe not. Again - it really depends on the photograph that you are making.
Color calibration is the process of calibrating disparate devices so that what's seen on one is seen on another, so that one can expect accuracy throughout the workflow. Generally, this means calibrating a monitor to display accurately and a printer to print what was displayed, accurately.