Dye ink, more thin and transparent,dries almost immediately on any surface. It's not good for any technique where the color solution needs to remain wet, even for a short period. Dye inksare not good for embossing, one-offs, or monoprinting. It also is not suitable for techniques where the solution has to be moved around, like smearing or smudging, etc.
Pigment ink, which is thicker and richer,takes longer to dry, especially on glossy, slick, coated paper, etc. This slow drying time allows for embossing powder to be applied and heat set. You can smear and smudge pigment inks, and do simple, quick mono-prints, though block printing ink is best for most one-offs and mono-printing.
How do I remember the difference? I made an association in my mind. To me the word dye means to stain, quickly and strongly, as if to dry quickly. It reminds me of being sorry that something made a stain and I can't get it off because it stained so deeply and richly and dried so fast. The word dye also reminds me of fabric dye, which is used to swiftly stain cloth a specific color. Swiftly relates to drying quickly. These associations help me to remember that pigment ink then is the slow-drying one.
PaperWishes.com has a good guide on understanding ink and stamp pads: Stamping 101 - About Ink Pads.
See this 5 min video at Youtube All About Stamping - Inks 101: Difference Between Dye, Pigment, Chalk, Etc. Inks , or watch it below.
The Creative Seller