Intelligence is used in random and non-random events, in combination with your encrypt/decrypt techs and open/closed sliders to determine your relative intelligence rating with another nation. If you have the higher rating, you might get a random event to steal blueprints from, sabotage them, and even get a surprise combat bonus when fighting them.
Army intelligence works pretty much the same way, adding on to your encr/decr techs and sliders, and is what determines whether or not you can see the number of divisions in an enemy province. If your rating is low, you might just see ?/?, if it's medium, something like 3?/6?, and if high the actual numbers, like 2/7.