They're often used in the same way -- i.e. to jot down lots of little ideas and organize projects or thoughts! The difference, functionally, is that post it notes are sticky, so you can affix them to a wall or table in a pattern; index cards aren't sticky, so while you can array them around a table you can't stick them in place on a wall.

On the other hand, index cards have useful attributes that post it notes don't; because they're not sticky (and typically a firmer type of cardboard) you can carry them around in a deck and sort of flip through them, which can be quite useful. And because they're not sticky, you can slide them around on a table, which can be a preferred way of viewing and organizing a lot of ideas ...