You can buy either waterproof or water-soluble inks. If you use waterproof ink, you can let it dry and then apply watercolor paint over it---a common technique used in illustration. Watercolors are always water-soluble.
While both can be applied with a brush, ink is also frequently applied with cartridge pens or drawing nibs. Nibs are pointed pieces of metal that hold a small amount of ink, acquired by dipping the nib in an ink bottle.
Watercolors are generally used for color washes without clearly defined lines. Inks can be watered down and used as washes, but they are most commonly used for line art.
Inks are generally used at full strength, while watercolors always have to be mixed with water. This makes it a bit easier to get consistent results with ink, as watercolors always involve some guesswork.
You can see marks on the paper beneath watercolors, even when they are applied in several layers. Ink is opaque unless it has been watered down for use as a wash.