I used to be a all-photoshop guy but nowadays I do mostly all of my stuff in illustrator.
#1: Logos
A logo should be able to scale to any format and should therefore be done in vectors. Photoshop really only have pseudo vectorabilities compared to Illustrator.
#2: Illustrations. I use illustrator to make most of my illustrations. Sometimes I’ll import it to Photoshop for finishing touches but mostly I start and end it in illustrator.
Examples of typical “Illustrator”-stuff which would be silly to do in PS instead of AI.
#3: Flash stuff.
Since Flash also uses vectors Illustrator is great for creating artwork for animations. An example can be seen here. The zooming animations in this site wouldn’t look as good with pixel graphics as it does with vectors.
That’s what I can think off right now. Once you get a taste for vectors you’ll never look back. 
-m