It's very much dependent on the type of graphic work that you're doing.
When using photoshop, you are creating raster graphics that are set in a certain amount of pixels (resolution), and are therefore not suited to enlarging due to 'pixellation'. It's perfectly suited to raster graphics like photos etc.
Illustrator on the other hand creates vector images, where the lines, shapes, fills and strokes are stored as a mathematical formula. For this reason, vector images are suited to creating graphics that could require resizing/scaling, as they don't pixellate and lose quality at any size. Typical uses would be logos.
For these reasons, I don't think that there is an 'either or' answer. For anyone that is serious about graphic design, both a vector and a raster graphics program should be part of your toolkit.