<p>When choosing which filament to use for a particular print does the color of the filament have any impact on it's performance, or is it purely a cosmetic choice?</p>
<p>For example, are there any side-by-side comparisons available that demonstrate that differences do\don't exist between different color filaments from the same company\range, such as a particular filament needing a hotter bed because the pigment changes it's properties?</p>

Yes in some cases it does.
It is widely known in communities which are pickier about properties that white filament requires a percentage of pigments much higher than other colours, therefore white filaments are typically weaker (or significantly weaker) than most other colours.
Another exception is black: ABS is not UV resistant, but if you use black ABS, the UV will not penetrate and, unless the surface is continuously scraped away, the UV resistance of black ABS is better than other colours.
I remember some mentions in the Voron discord chats that red (at least from some manufacturers) is better than some other colours from the same manufacturer, but I don't remember the details. You can
join
the chat yourself and search (before asking!) in the #filament chat.
I remember reading that pure ("natural") filaments are however not always better than filaments with colours added, but I cannot remember the source. It could be that some little amount of extra additives (the colour pigments) offer the polymer something to "grab", so better than nothing. My guess, but I have no support for it.