Generally, it's the pigment size, it's much smaller in inks as they need to fit through a pen tip. It may not be much smaller compared to the super fine pigments in some of the higher end airbrush paints, I've never compared the data.
Inks used to be dye based colours (some still are) but pigments seem to be taking over these days.
I think inks are used when you need a more fluid medium and/or a more saturated colour. They're also popular to deepen/brighten/increase contrast/etc.
You've probably seen miniatures painted white then layered with yellow ink to give a really nice, rich yellow armour, Eldar and SM's do that sort of thing all the time. This works due to their transparency, it works well with browns and blacks for fur too. It does most of your shade, mid, highlight work all in one go. It wouldn't work the same on a large flat panel/canvas but...yeah.