I can't give you a detailed description about why separating colors in Photoshop must to be fundamentally different than separating colors in Illustrator. But the difference can be summarized by pointing out: Photoshop color separation is channel-based. Illustrator color separation is swatch based.
It's much simpler in Illustrator: for each color you use, double click on its swatch. Change the color from a "process color" to a "spot color."
When you want to print the separations, go to "print," select "output," and change the mode from "composite" to "separations." You can control which colors to print by clicking on the print icon next to the swatch/channel.
Don't forget your registration marks (under "Marks & Bleed")!